Amplification of Completely Bounded Operators \\ and Tomiyama's Slice Maps
Matthias Neufang
Let $(\em,\en)$ be a pair of von Neumann algebras, or %$\sigma$-weakly closed subspace of of dual operator spaces with at least one of them having property $S_\sigma$, and let $\Phi$ be an arbitrary completely bounded mapping on $\em$. %and %let $\Phi$ be an arbitrary completely bounded mapping on $\em$. We\footnote{2000 {Mathematics Subject Classification}: %03E55, 22D05, 46B10, 46B26, 46L10. 46L06, 46L07, 46L10, 47L10, 47L25. \par {Key words and phrases}: completely bounded operator, amplification, Tomiyama's slice maps, von Neumann algebra, dual operator space, %operator approximation property $S_\sigma$, projective operator space tensor product. %tom's sl maps am ende? \par The author is currently a PIMS Postdoctoral Fellow at the %This work was accomplished while the author was a PIMS Postdoctoral Fellow at the University of Alberta, Edmonton, where this work was accomplished. The support of PIMS is gratefully acknowledged.} %show the existence and even present an explicit construction of an amplification of %an arbitrary completely bounded mapping $\Phi$ %on $\em$ to a completely bounded mapping on $\em \en$. %, where $\en$ is either another von Neumann algebra or %an arbitrary dual operator space, respectively. Our approach is based on the concept of slice maps as introduced by Tomiyama, and makes use of the description of the predual of $\em \en$ given by Effros and Ruan %in in terms of the operator space projective tensor product (cf.\ \cite{efru}, \cite{ru}). \par We further discuss %the uniqueness of such an amplification as well as various several properties of %this general amplification an amplification in connection with the investigations made in \cite{mnw}, where the special case $\em=\B(\H)$ and $\en=\B(\Ka)$ has been considered (for Hilbert spaces $\H$ and $\K$). We will then %present mainly focus on various applications, such as a remarkable purely algebraic characterization of $w^*$-continuity using amplifications, as well as a generalization of the so-called Ge--Kadison Lemma (in connection with the uniqueness problem of amplifications). Finally, our study %In particular, this will enable us to show that the essential assertion of the main result in \cite{mnw} concerning completely bounded bimodule homomorphisms actually relies on %a fundamental and %an elementary a basic property of Tomiyama's slice maps. % %thus providing a new approach to %the latter %this result.
math0201078.tex
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%%%%IRGENDWO 
%%%%%%%%%%ACHTUNG: zwei Bindestriche zwischen mehrautoren!!!!!! 
%%wohl am anfang nicht unbedingt erwaehnen, dass kraus fuer duale ORe deCann-Haag gemacht hat!?, eo
%%%%STATT S_sigma natuerlicher: aequivalenz v effr-ruan in approxprop's, eventuell
%%%%auch alle moeglen angeben, auch mit kraus; u zushang zu eind-kriterien, vgl auch Tomiyama's untersuchungen
%%%%bei darsten!!!??, eeeooo
%%%%%%%%wie mit eindkeit bei dualen ORen mit irgendwelchen eigschaften!?: Ssigma, eo
%%%%besser statt Ssigma appox-eig u eff-rua-kraus' 1000 aequivalenzen erwaehnen!?, eeeee oo
%%sagen: bei dualem OR stets von fixiertem praedual spreche ich; dh:
%%dualer OR hat einen praedual, u wir fixieren mit \em gleich einen 1 fuer allemal!!!
%%%%%%%%sagen: auch im vgl zu effr-ruan: bei mir als 2. vna wichtig bzw noch allg;
%%%%%%%%braucht Tomitas Satz in tp_proj-resultat (mit Tomitas Satz)---> so: explizit!
%%%%%%%%witzig: wie bei eind 1. (!) braucht einschr fuer belen 2.en - aber wohl wegen symm des tp doch egal!?, eeeooo : !!??
%%%manchmal eher Phi(u), ... statt 3 tensoren rechts schreiben !?
%%das auch irgendwie multilinear !?, eo; vgl mit effros-exel-kish!?, hofm, eeeooo
%%%%%%VERTAUSCHUNGSFORMEL LINKS U RECHTS RAUSSTELLEN, die auch
%%%%%%aehnlich bei kraus steht?!, eo
%%%%%%%%%%SAGEN, dass wir erstmalig bezug ampl u slice maps herstellen!?, eo
%%%s.u. mit projTP=nuc!? - vgl effr-kraus-rua!?, eo
%%%Charisierung der avN mit S_s, nicht allg duale ORe!?, eo
%\em inj faktor --> semidiskret --> (Kraus) hat S_sigma
%%%%%Tomiyama-Vorl zur Nicht-Eindkeit eo erwaehnen!
%%%%admissible pair einfuehren!!
%%%%%%%AUCH UNSICHER, OB ALL DIE EIGschaften eo U/O EINDKEIT EO GELTEN u/o
%%%%%%%einfach zu beweisen mit
%%%%%%%Stratila-approach!?, eo
%%%%%%%%%%%%SAGEN WIE ANSELM: TOLL, DASS DAS MEISTE AUS DER 1 DEFinitorischen GLEichung fliesst (vgl. Eric!!, formel!),
%%%%%%%%%%%%UND ALLES REIN ALGEBRAISCH , EINFACH [GENIAL], ausser vollst isometzrie, eo: da OR-Theorie!?, eo
%%%%%%%%%AUCH SAGEN: STRATILA OHNE OR-Theorie gut, HEUTE NEUER=MEIN ZUGANG!!!?, eeeeeeeo
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%%%%%%%%%%die benutzen und aber nicht(!!!!!!!!!!!!): fuer p=2 das alte, da das in Cbbh dargestellt war!! !!!!!!!!!!!!!
%%%%auch beachten: bei meinen modulhom's: habe ich notation eingefuehrt mit "unten l, L_1" eo !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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%echt richtig?, eo: vgl im eingebundenen tom-tex-Kap!?, eo
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%oder doch mit Z_{mathrm t} schreiben, eeeeeooooo !?
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%
%\begin{center}
%{\Large{\textsc{On Mazur's property and property $(X)$}}}
%\end{center}
%
%\begin{titlepage}
\title{Amplification of Completely Bounded Operators \\ and
Tomiyama's Slice Maps}
\author{Matthias Neufang}
%\thanks{The author is currently a PIMS Postdoctoral Fellow at the
%This work was accomplished while the author was a PIMS Postdoctoral Fellow at the
%University of Alberta, Edmonton, where this work was accomplished. The support of PIMS is gratefully acknowledged.}}
%???: \keywords{...}
\date{}
\maketitle
\begin{abstract}
Let $(\em,\en)$ be a pair of von Neumann algebras, or 
%$\sigma$-weakly closed subspace of
of dual operator spaces with 
at least 
one of them having property $S_\sigma$, and let $\Phi$ be an arbitrary completely bounded mapping on $\em$. 
%and 
%let $\Phi$ be an arbitrary completely bounded mapping on $\em$.
We\footnote{2000 {\textit{Mathematics Subject Classification}}:
%03E55, 22D05, 46B10, 46B26, 46L10.
46L06, 46L07, 46L10, 47L10, 47L25.
\par
{\textit{Key words and phrases}}: completely bounded operator, amplification,
Tomiyama's slice maps,
von Neumann algebra, dual operator space,
%operator approximation
property $S_\sigma$, projective operator space
tensor product.
%tom's sl maps am ende?
\par
The author is currently a PIMS Postdoctoral Fellow at the
%This work was accomplished while the author was a PIMS Postdoctoral Fellow at the
University of Alberta, Edmonton, where this work was accomplished. The support of PIMS is gratefully acknowledged.}
%show the existence and even 
present an explicit construction of an amplification of
%an arbitrary completely bounded mapping
$\Phi$ 
%on $\em$ 
to a completely bounded mapping on $\em \tt \en$. 
%, where $\en$ is either another von Neumann algebra or
%an arbitrary dual operator space, respectively.
Our approach is based on the concept of slice maps as introduced by Tomiyama, and
makes use of the description of the predual of $\em \tt \en$ given by Effros and Ruan
%in
in terms of the operator space projective tensor product (cf.\ \cite{efru}, \cite{ru}).
\par
We further discuss 
%the uniqueness of such an amplification as well as various
several properties of 
%this general amplification
an amplification in connection with the investigations made in \cite{mnw}, where the special case
$\em=\B(\H)$ and $\en=\B(\Ka)$ has been considered (for Hilbert spaces $\H$ and $\K$). 
We will then 
%present 
mainly focus on various applications, such as 
a remarkable purely \textit{algebraic} characterization of $w^*$-continuity using amplifications, as well as 
a generalization of the so-called Ge--Kadison Lemma (in connection with the uniqueness problem of amplifications). 
Finally, our study 
%In particular, this 
will enable us
to show that the essential assertion of the main result in \cite{mnw} concerning completely bounded bimodule 
homomorphisms 
actually relies on
%a fundamental and
%an elementary
a basic property of Tomiyama's slice maps.
%
%thus providing a new approach to
%the latter
%this result.
\end{abstract}

\section[Introduction]{Introduction}
\noindent
The initial aim of 
%the present 
this article
is to 
%prove the existence and, at the same time, to
present an \textit{explicit} construction of
%construct explicitly
the amplification
of an arbitrary completely bounded mapping on a von Neumann algebra $\em$ to a completely bounded mapping
%zu einer vollst\"andig beschr\"ankten Abbildung auf
on $\em \tt \en$,
%zu amplifizieren, wobei
where $\en$ denotes another von Neumann algebra, and to give various applications. 
%wie auf C*-Algebren!?
\par 
%This generalizes the known result on the existence of an amplification of
Of course, an amplification of 
%Dies verallgemeinert das bekannte Resultat \"uber die Existenz einer Amplifikation von
%{\textit{normal}}
a completely bounded mapping on a von Neumann algebra 
is easily obtained if the latter is assumed to be {\textit{normal}} (i.e., $w^*$-$w^*$-continuous); 
cf., e.g., \cite{haa}, Lemma 1.5 (b). Our point is that we are dealing with not necessarily normal mappings 
and nevertheless even come up with an explicit formula for an amplification. 
\par 
We further
remark
%stress
that, of course,
%the result is
as already established by Tomiyama, it is possible to amplify {\textit{completely positive}} mappings on
von Neumann algebras (\cite{strat}, Prop.\ 9.4). One could then think of Wittstock's decomposition
theorem to write an arbitrary completely bounded mapping on the von Neumann algebra $\em$
as a sum of four completely positive ones and apply
Tomiyama's result -- but the use of the decomposition theorem requires
%the injectivity of
$\em$ to be injective.
%Furthermore,
Moreover, we wish to stress that this procedure would be highly non-constructive:
first, the proof of Tomiyama's result uses Banach limits in an essential way, and the decomposition theorem as well
is an abstract existence result.
%ausserdem sagen: verallg auf duale ORe
%involving the hbth for cp/cb maps!?, eo
\par
We even go on further
%and
%prove
to show that the same construction can actually be carried out in case $\em$
%is a dual operator space with
%property $S_\sigma$ and $\en$
and $\en$ are only required to be dual operator spaces, where at least one of them shares property $S_\sigma$ (the latter
is satisfied, e.g., by all semidiscrete von Neumann algebras). In this abstract situation, we do not even have
a version of the above mentioned result of Tomiyama at hand.
%u witt-satz!?, eo
%
%(if $\em$ is a
%von Neumann algebra, this latter property is satisfied).
\par
%We further remark that 
Furthermore, our approach yields, in particular,
a new and elegant construction of the usual amplification
%liefern
%wir im Falle, da"s
%WIE IMMMER MIT EIND.EN!!????
%$\em=\B(\H)$ und $\en=\B(\Ka)$, wobei $\H$ und $\Ka$ Hilbertr\"aume sind,
%eine vollkommen neue und elegante Konstruktion der \"ublichen Amplifikation
%das kann wmn schwer, nur mit fortssatz eo auf bel vna hochziehen, auch dort darst von normalen von haag gebraucht, oder wie?
%und oder dichtesaetze a la chatterjee-smith-hofm !?
of an operator $\Phi \in \C\B(\B(\H))$ to an operator in $\C\B(\B(\H \otimes_2 \Ka)) = \C\B(\B(\H) \tt \B(\Ka))$,
i.e., of the mapping
%of
$$\Phi \mapsto \Phi^{(\infty)}$$
%die
%an welche wir
%in Proposition \ref{hallos} erinnerten.
%considered
introduced in \cite{efki} (p.\ 265) and studied in detail in \cite{mnw} (cf.\ also \cite{bim}, p.\ 151 and Thm.\ 4.2, where
instead of $\B(\H)$, an arbitrary dual operator space is considered as the first factor).
But of course, the main interest of our approach lies
in the fact that it yields an explicit and constructive description of the amplification in a much more general setting -- where
the second factor $\B(\Ka)$ is replaced by an arbitrary von Neumann algebra or even a
%n arbitrary
dual operator space. In this case, the very definition of the mapping $\Phi \mapsto \Phi^{(\infty)}$ does not make sense,
since it is based on the representation of the elements in $\B(\H) \tt \B(\Ka)$ as infinite matrices with entries from $\B(\H)$.
%%%Over and above that,
\par
The crucial idea in our construction is to use the concept of slice maps as introduced by Tomiyama,
in connection with the explicit description of the predual $(\em \tt \en)_*$,
%in terms of the
%projective operator space tensor product,
given by Effros and Ruan --
cf.\
%in
\cite{efru}, for the case of von Neumann algebras, and \cite{ru}, for dual operator spaces with the first factor enjoying
property $S_\sigma$. In these two cases, one has a canonical complete isometry
\begin{eqnarray}
(\em \tt \en)_* \cbgl \em_* \tp \en_*, \label{projtp}
\end{eqnarray}
where the latter denotes the projective operator space tensor product.
The dichotomic nature of our statements precisely arises from this fact; at this point, we wish to emphasize that,
as shown in \cite{ru}, Cor.\ 3.7, for a dual operator space $\em$, (\ref{projtp}) holding for all dual
operator spaces $\en$, is actually equivalent to $\em$ having property $S_\sigma$.
%%--> implikationen bei mir!?, eo
%%%%andere aequivalente bed.en a la effros-kraus-ruan u a unveroefflichte sachen, eo, runde fragen, eeeooo, auch
%%%%von witt, eo: bimoduln, kraus-ruan: EEEEEOOOOO
%
%the fact that
%%%%%%%%%%%%%%%%%%steht wmn-ich-bem oben an richtiger stelle!?, eeoo
\par
%So,
Hence, the
combination of operator space theory and the classical operator algebraic tool provided by
%the
Tomiyama's slice maps enables us to show that the latter actually encodes all the
essential
%crucial
properties
%ty
of an amplification.
Summarizing the major advantages of our approach to the amplification problem, we
%finally
stress the following:
\begin{itemize}
%\item[(a)]
%The
%combination of operator space theory and the classical operator algebraic tool provided by
%the
%Tomiyama's slice maps enables us to show that the latter actually encodes all the
%essential
%crucial
%properties
%ty
%of an amplification.
\item[(a)]
The use of Tomiyama's slice maps gives rise to
%a simple
an explicit formula for the amplification of arbitrary
completely bounded mappings with a simple structure.
\item[(b)]
In particular, the approach is constructive.
\item[(c)]
In various cases, it is easier to handle amplifications using our formula,
since the construction does not involve any ($w^*$-)limits.
As we shall see,
%the manipulations required in
the investigation of the amplification mapping
is thus mainly reduced to purely algebraic considerations -- in
contrast to the somehow delicate analysis used
in \cite{bim} or \cite{mnw}.
\item[(d)]
Our framework -- especially concerning the class of
objects allowed as ``second factors'' of amplifications -- is by far more general
than what can be found in the literature.
\end{itemize}
We would finally like to point out that,
in view of the above mentioned characterization of dual operator spaces fulfilling
equation (\ref{projtp}), via property $S_\sigma$ (\cite{ru}, Cor.\ 3.7),
our approach of the amplification problem
seems ``best possible'' among the approaches satisfying (a)--(d).
%Thus, the combination of operator space theory and the classical operator algebraic tool provided by
%the
%Tomiyama's slice maps enables us to show that the latter actually encodes all the
%essential
%crucial
%properties
%ty
%of an amplification, and even gives rise to a simple formula for the amplification of arbitrary
%completely bounded mappings. A major advantage of this approach consists
%in the fact that, as we shall see, the manipulations required in the investigation of the amplification mapping
%are mainly algebraic -- in contrast to the somehow delicate analysis involved in the work \cite{mnw} or \cite{bim}.
%%nachschauen!!: macht er das echt!?, eo : stratile gebraucht auch Tom's sl maps!?,
%%aber wieso dann nicht explizite konstr!?
\par
The paper is organized as follows. -- First, we
%present
provide the necessary terminological background from the theory of
operator algebras and
operator spaces. We further give the exact definition of what we understand by an ``amplification".
The construction of the latter in our general situation is presented in section \ref{mainconst}. 
%where also a uniqueness result is obtained.
\par
Section \ref{disc} contains a discussion of this amplification mapping under various aspects, e.g.,
relating it to results from \cite{haa} and \cite{mnw}. We derive a stability result
%with respect to
concerning the passage to von Neumann subalgebras or dual operator subspaces, respectively. Furthermore,
an alternative (non-constructive) description of our amplification mapping is given,
which, in particular, immediately implies that the latter preserves complete positivity.
%from which it is imm seen ...
%is
\par
In section \ref{tow}, we
introduce an analogous concept of ``amplification in the first variable" which leads to an interesting comparison with the
usual tensor product of operators on Hilbert spaces. The new phenomenon we encounter here is that the amplifications, one in 
the first, the other in the second variable, of two completely bounded mappings $\Phi \in \C\B(\em)$ and 
$\Psi \in \C\B(\en)$ do \textit{not} commute in general. They do, though, if $\Phi$ or $\Psi$ happens to be normal (cf.\ Theorem 
\ref{commute} below). 
%Our striking result 
But our main result of this section, which may even seem a little estonishing, states that for most von Neumann algebras 
$\em$ and $\en$, the converse holds true. This means that 
the apparently weak -- and purely algebraic! -- 
condition of commutation 
%for the associated 
of the amplification of $\Phi$ with every amplified $\Psi$ already suffices to ensure that $\Phi$ is normal (Theorem \ref{umk}). 
%briefly compare
%In particular, we derive (under some necessary condition) a commutation relation
%satisfied by the two different kinds of amplification.
The proof of the latter assertion is rather technical and involves the use of countable spiral nebulas, a concept first 
introduced by Hofmeier--Wittstock in \cite{wiho}. 
\par 
In the section thereafter, 
%is devoted 
we turn to the uniqueness problem of amplifications. Of course, an amplification which only 
meets the obvious algebraic condition is far from being unique; but we shall show that 
%in two 
with some (weak) natural assumptions, one can indeed establish uniqueness. One of our two positive results 
actually yields a considerable generalization (Proposition \ref{ichgekad}) of a theorem which has proved useful in the solution of the 
splitting problem for von Neumann algebras (cf.\ \cite{gekad}, \cite{stzs}), and which is sometimes referred to as the 
Ge--Kadison Lemma. 
\par 
Finally, section \ref{sechs} is devoted to
%present
a completely new approach to a theorem of May, Neuhardt and Wittstock regarding a
module homomorphism property of the amplified mapping. We prove a version of their result in our
context, which reveals that the essential assertion in fact relies on a fundamental and elementary
module homomorphism property shared by Tomiyama's slice maps.
%is devoted to
%
%%%%%%%OBEN AUCH NOCH KOMPTAET ERWAEHNEN, DANN VGL MIT
%%%%%%%klassischen TP von Hilbraum-Open,
%%%%%%%dann WMN-Beweis!?, eo

\section{Preliminaries and basic definitions}
%
%
%etwa in
%\cite{efki} (insbesondere S.\ 265)
%oder
%und \cite{mnw}
%(s.\ \S 2, S.\ 284) studiert wird.
\par
%for
%and
%$\Ka$ are Hilbert spaces.
Since
%several
%sources of
a detailed
%expositions
account of the theory of operator spaces, as developed by Blecher--Paulsen, Effros--Ruan, Pisier \textit{et al.}, 
%are
is by now available via different sources (\cite{efruan}, \cite{pis}, \cite{witt}),
as for its basic facts, we shall restrict ourselves to fix the terminology and notation we use. 
\par 
Let $X$ and $Y$ be operator spaces. We denote by $\C\B(X, Y)$ the operator space of completely bounded maps
from $X$ to $Y$, endowed with the completely bounded norm $\| \cdot \|_{\mathrm{cb}}$. We further write
$\C\B(X)$ for $\C\B(X, X)$.
If $\em$ and $\rr$ are von Neumann algebras with $\rr \subseteq \em$, then we
denote by $\C\B_{\rr} (\em)$ the space of all completely bounded
$\rr$-bimodule homomorphisms on $\em$.
%Given a dual operator space $X^*$, we denote by $\C\B^{\sigma} (X^*)$ the
%subspace of $\C\B(X^*)$ consisting of normal (i.e., $\sigma(X^* , X)$-$\sigma(X^*, X)$-continuous) mappings. 
If 
two operator spaces 
$X$ and $Y$ are completely isometric, then we write $X \cbgl Y$. 
\par 
An operator space $Y$ is called a \textit{dual operator space} if there is an operator space $X$ such that $Y$ is
completely isometric to $X^*$. In this case, $X$ is called an \textit{operator predual}, or \textit{predual}, for short, of
the operator space $Y$. In general, the predual of a dual operator space is not unique (up to complete isometry);
see \cite{ru}, p.\ 180, for an easy example.
%%Auch: = erklaeren!!: was genau Bedeutung!!
%\par
%In
%the following
%this article, whenever we are speaking about ``a dual operator space", we always mean ``a dual operator space $\em$ with a given
%(operator) predual, denoted by $\em_*$".
\par
For a dual operator space $Y$ with a given predual $X$, we denote by $\C\B^{\sigma} (Y)$ the
subspace of $\C\B(Y)$ consisting of normal (i.e., $\sigma(Y , X)$-$\sigma(Y , X)$-continuous) mappings.
\par
If $\H$ is a Hilbert space, then, of course, every $w^*$-closed subspace $Y$ of $\B(\H)$ is a dual operator space with a
canonical predual $\B(\H)_{*}/Y_{\perp}$, where $Y_{\perp}$ denotes the preannihilator of $Y$ in $\B(\H)_*$. We denote
this canonical predual by $Y_*$.
Conversely,
if
%for every
$Y$ is a
dual operator space with a given predual $X$, there is a $w^*$-homeomorphic complete isometry of $Y$ onto a $w^*$-closed
subspace $\widetilde{Y}$ of $\B(\H)$, for some suitable Hilbert space $\H$ (see \cite{efru}, Prop.\ 5.1). Following the
terminology of \cite{efkrru}, p.\ 3, we shall call such a map a $w^*$-\textit{embedding}. In this case, $X$ is completely isometric
to the canonical predual $\widetilde{Y}_*$, and we shall identify $Y$ with $\widetilde{Y}$
%and still write $Y_*$ instead of $X$
%%for this, cf. ...
(cf.\ the discussion in
\cite{ru}, p.\ 180).
\par
We denote by $\H \otimes_2 \Ka$ the Hilbert space tensor product of two Hilbert spaces $\H$ and $\Ka$.
For operator spaces $X$ and $Y$, we write $X \ti Y$ and $X \tp Y$ for the
injective and projective operator space tensor product,
%of operator
%spaces $X$ and $Y$,
respectively.
We recall that, for operator spaces $X$ and $Y$, we have a canonical complete isometry:
$$(X \tp Y)^* \cbgl \C\B(X, Y^*).$$
%\par
If $\em$ and $\en$ are von Neumann algebras, we denote as usual by $\em \tt \en$ the von Neumann algebra tensor product.
%\par
More generally, let $V^*$ and $W^*$ be dual operator spaces with given $w^*$-embeddings $V^* \subseteq \B(\H)$ and
$W^* \subseteq \B(\K)$. Then the $w^*$-spatial tensor product of $V^*$ and $W^*$, still denoted by
$V^* \tt W^*$, is defined to be the $w^*$-closure of the algebraic tensor product $V^* \otimes W^*$ in $\B(\H \otimes_2 \Ka)$.
The corresponding $w^*$-embedding of $V^* \tt W^*$ determines a predual
%which we denote by
$(V^* \tt W^*)_*$.
Since we have
$$(V^* \tt W^*)_* \cbgl V \otimes_{\mathrm{nuc}} W$$
%holds
completely isometrically (where $\otimes_{\mathrm{nuc}}$ denotes the nuclear operator space tensor product),
the $w^*$-spatial tensor product of dual operator spaces does not depend on the given $w^*$-embedding (cf.\ \cite{efkrru}, p.\ 127).
%%Phi_infty erklaeren!!
%%aequivalenzen d eigschaften angeben, semid hat S_sigma
%%slice maps fuer dual op spaces!!
\par
In
the following,
%this article,
whenever we are speaking of ``a dual operator space", we always mean ``a dual operator space $\em$ with a given
(operator) predual, denoted by $\em_*$".
\par
We will now present, in the general framework of dual operator spaces, the construction of left and right slice maps, following
the original concept first introduced by Tomiyama for von Neumann algebras (\cite{tom}, p.\ 4;
cf.\ also \cite{kraus}, p.\ 119).
%\S ... sagen!?
-- Let $\em$ and $\en$ be dual operator spaces. For any $\tau \in \en_*$ and any $u \in \em \tt \en$, the mapping
$$\em_* \ni \rho \mapsto \langle u, \rho \otimes \tau  \rangle$$
is a continuous linear functional on $\em_*$, hence defines an element $L_\tau (u)$ of $\em$.
%It
As is
%now
easily verified,
%that
$L_\tau$ is the unique $w^*$-continuous linear map from $\em \tt \en$ to $\em$ such that
$$L_{\tau}(S \otimes T) = \langle \tau, T \rangle ~S \quad (S \in \em, ~T \in \en);$$
it will be called the left slice map associated with $\tau$. In a completely analogous fashion, we see that for every
$\rho \in \em_*$, there is a unique $w^*$-continuous linear map $R_\rho$ from $\em \tt \en$ to $\en$ such that
$$R_{\rho}(S \otimes T) = \langle \rho, S \rangle ~T \quad (S \in \em, ~T \in \en),$$
%it will be
which we call the right slice map associated with $\rho$.
\par
We note that, as an immediate consequence, we have for every $\rho \in \em_*$, $\tau \in \en_*$ and
$u \in \em \tt \en$:
$$\langle L_\tau (u), \rho \rangle = \langle u, \rho \otimes \tau \rangle \quad \mathrm{and}
\quad \langle R_\rho (u), \tau \rangle = \langle u, \rho \otimes \tau \rangle.$$
%\par
%We
Let us now briefly recall the definition of property $S_\sigma$ for dual operator spaces, as introduced by Kraus
%in the von Neumann algebra case
(\cite{krtrans}, Def.\ 1.4). To this end, consider two dual operator spaces
$\em$ and $\en$ with $w^*$-embeddings $\em \subseteq \B(\H)$ and $\en \subseteq \B(\K)$.
%and by Ruan for arbitrary
Then the Fubini product of $\em$ and $\en$ with respect to $\B(\H)$ and $\B(\K)$ is defined through
$$\mathcal{F} (\em , \en , \B(\H) , \B(\K)) \dogl \{  u \in \B(\H \otimes_2 \K) \mid L_\tau (u) \in \em ,
~R_\rho (u) \in \en ~\mathrm{for ~all}~ \rho \in \B(\H)_* , ~\tau \in \B(\K)_* \} .$$
%for all $\rho \in \B(\H)_* = \mathcal{T} (\H)$ and $\tau \in \B(\K)_* = \mathcal{T} (\K)$.
By the important result \cite{ru}, Prop.\ 3.3, we have
$$\mathcal{F} (\em , \en , \B(\H) , \B(\K)) \cbgl ( \em_* \widehat{\otimes} \en_* )^* ,$$
which in particular shows that the Fubini product actually does not depend on the particular choice of Hilbert spaces $\H$ and $\K$
(cf.\ also \cite{krtrans}, Remark 1.2).
Hence, we may denote the Fubini product simply by $\mathcal{F} (\em , \en)$. Now, a dual operator space $\em$
is said to have property $S_\sigma$ if
$$\mathcal{F} (\em , \en) \cbgl \em \tt \en$$
holds for every dual operator space $\en$.
\par
The question of whether every von Neumann algebra has property $S_\sigma$ has been answered in the
negative by Kraus (\cite{kraus}, Thm.\ 3.3); in fact, there exist even separably acting factors without property $S_\sigma$
(\cite{kraus}, Thm.\ 3.12).
\par
But every
%To give a concrete example, we only mention that every
semidiscrete von Neumann algebra -- and hence, every type I von Neumann algebra (\cite{efla}, Prop.\ 3.5) --
has property $S_\sigma$, as shown by Kraus in \cite{krtrans}, Thm.\ 1.9.
We finally
%mention
refer to the very interesting results
%that
%As shown by
obtained by Effros, Kraus and Ruan
%,
%prove the equivalence of
showing that property $S_\sigma$
is actually equivalent
to various operator space
approximation properties (cf.\ \cite{kraus}, Thm.\ 2.6; \cite{efkrru}, Thm.\ 2.4).
\par
To avoid long paraphrasing, let us
%first
now
introduce some terminology
suitable for the statement of our results.
\begin{defi} \label{pair}
Let $\em$ and $\en$ be either
\begin{itemize}
\item[(i)] arbitrary von Neumann algebras,
\item[or]
\item[(ii)] dual operator spaces such that at least one of them has
%the OAP.
property $S_\sigma$.
\end{itemize}
Then, in both cases, we call ($\em$, $\en$) an
${\mathrm{admissible ~pair}}$.
\end{defi}
%
%
%%If ($\em$, $\en$) is an admissible pair, then, owing to \cite{ru}, Cor.\ 3.6
%(or )
%%and Cor.\ 3.7, respectively, we have
%the following
%%a canonical
%description of the predual of $\em \tt \en$
%%complete isometry:
%%$$(\em \tt \en)_* \cbgl \em_* \tp \en_*.$$
%
%%%%%%EINFACHER zu sehen, dass semidiskrete vna's S_sigma haben????
%\par
%%As already pointed out, property $S_\sigma$ of a dual operator space $\em$ is actually characterized by the above equality
%%holding for all dual operator spaces $\en$ (\cite{ru}, Cor.\ 3.7).
If ($\em$, $\en$) is an admissible pair, then 
%(or )
%and Cor.\ 3.7, respectively, 
we have 
%the following 
a canonical 
%description of the predual of $\em \tt \en$
complete isometry: 
$$(\em \tt \en)_* \cbgl \em_* \tp \en_*.$$ 
%
%%%%%%EINFACHER zu sehen, dass semidiskrete vna's S_sigma haben???? 
%\par 
In case $\em$ and $\en$ are both von Neumann algebras 
or dual operator spaces with $\em$ having property $S_\sigma$, 
this follows from \cite{ru}, Cor.\ 3.6 and 
Cor.\ 3.7, respectively. 
%(As already pointed out, property $S_\sigma$ of a dual operator space 
%$\em$ is actually characterized by the above equality 
%holding for all dual operator spaces $\en$; see \cite{ru}, Cor.\ 3.7.) 
\par 
If $\em$ and $\en$ are dual operator spaces with $\en$ enjoying 
property $S_\sigma$, the above identification is seen as follows. (We 
remark that the argument below, together with \cite{ru}, Cor.\ 3.7, 
shows that 
property $S_\sigma$ of a dual operator space 
$\em$ -- resp.\ $\en$ -- is actually characterized by the equality 
$(\em \tt \en)_* \cbgl \em_* \tp \en_*$ 
holding for all dual operator spaces $\en$ -- resp.\ $\em$.) 
\par 
Let $V$ be an arbitrary operator space. 
Then, by \cite{kraus}, Thm.\ 2.6, $V^*$ has property $S_{\sigma}$ if and only if it has the 
weak${}^*$ operator approximation property (in the terminology of \cite{efkrru}). Owing to \cite{efkrru}, Thm.\ 2.4 (6), 
this is equivalent to $$V \widehat{\otimes} W \cbgl V \otimes_{\mathrm{nuc}} W$$ holding for all operator spaces $W$. 
But since both the projective and the injective operator space tensor product are symmetric, 
this in turn is equivalent to the canonical mapping $$\Phi : W \widehat{\otimes} V \longrightarrow W \stackrel{\vee}{\otimes} V$$ 
being one-to-one for all operator spaces $W$. But this is the case if and only if we have 
$$W \otimes_{\mathrm{nuc}} V \cbgl W \widehat{\otimes} V$$ for all operator spaces $W$, which finally is of course equivalent to 
$$( W^* \tt V^* )_{*} \cbgl W \widehat{\otimes} V$$ holding for all operator spaces $W$. 
\par
We
%begin by making
shall now make precise what we
%shall
mean by an amplification, in particular, which
topological requirements it should meet beyond the obvious algebraic ones.
%unter einer Amplifikation verstehen, und betonen insbesondere, welche topologischen Eigenschaften
%eine Abbildung
%stellen zusammen
%wir
%von einer Amplifikation
%\"uber die offensichtlichen algebraischen hinaus verlangen.
\begin{defi} \label{amdef}
Let ($\em$, $\en$) be
%von Neumann algebras
an admissible pair. A
%stetige
completely bounded linear mapping $$\Chi: \C\B(\em) \longrightarrow \C\B(\em \tt \en),$$
%welche f\"ur alle $\Phi \in \C\B(\em)$, $S \in \em$, $T \in \en$ die
%Identit\"at
satisfying the
algebraic amplification condition ${\mathrm{(AAC)}}$
$$\Chi(\Phi)(S \otp T) = \Phi(S) \otp T$$
for all $\Phi \in \C\B(\em)$, $S \in \em$, $T \in \en$,
will be called
%hei"st
an ${\mathrm{amplification}}$, if in addition it enjoys the following properties:
\begin{itemize}
\item[(i)]
$\Chi$ is a complete isometry,
\item[(ii)]
$\Chi$ is multiplicative (hence, by (i), an isometric algebra isomorphism onto the image),
%also isomorphism into
\item[(iii)]
$\Chi$ is $w^*$-$w^*$-continuous,
\item[(iv)]
$\Chi(\C\B^{\sigma}(\em)) \subseteq \C\B^{\sigma}(\em \tt \en)$, i.e., normality of the mapping is preserved.
\end{itemize}
\end{defi}
\begin{bem}
In particular, $\Chi$ is unital. For $\Chi({\mathrm{id}}_{\em})$ coincides with
${\mathrm{id}}_{\em \tt \en}$ on all elementary tensors $S \otp T \in \em \tt \en$
due to ${\mathrm{(AAC)}}$; furthermore, by (iv), it is normal,
whence we obtain
%allen Elementartensoren $S \otp T \in \em \tt \en$ mit ${\mathrm{id}}_{\em \tt \en}$ \"uberein und ist wegen (iv) normal, so da"s
%bereits
$\Chi({\mathrm{id}}_{\em}) = {\mathrm{id}}_{\em \tt \en}$ on the whole space $\em \tt \en$.
\end{bem}
In \cite{mnw} -- cf.\ also \cite{efki}, p.\ 265 --,
%an amplification
for every $\Phi \in \C\B(\B(\H))$, the authors obtain a mapping
$\Phi^{(\infty)} \in \C\B(\B(\H \otimes_2 \Ka)) = \C\B(\B(\H) \tt \B(\Ka))$
which obviously fulfills the algebraic amplification condition (AAC). The definition of $\Phi^{(\infty)}$ is carried out in
detail in \cite{mnw}, p.\ 284, \S 2 (there, the Hilbert space $\Ka$ is written in the form $\ell_2 (I)$ for some
suitable set $I$). We restrict ourselves to recall that, given a representation
of $u \in \B(\H \otimes_2 \Ka)$ by an infinite matrix $[u_{i , j}]_{i , j}$, where $u_{i , j} \in \B(\H)$, one has:
$$\Phi^{(\infty)} ([u_{i , j}]_{i , j}) = [\Phi (u_{i , j})]_{i , j} .$$
%The following result is well-known.
In fact, as is well-known, more than only (AAC) is satisfied. Indeed, we have
\begin{satz} \label{wmnek}
The mapping
\begin{eqnarray*}
\C\B(\B(\H)) &\longrightarrow& \C\B(\B(\H) \tt \B(\Ka)) \\
\Phi &\mapsto& \Phi^{(\infty)}
\end{eqnarray*}
fulfills the above requirements of an amplification.
%gen\"ugt nach Proposition \ref{hallos} der obigen Definition einer
%Amplifikation.
%auch eff-kis u/o Eff-Exe vglen!?
%%den auch bringen in arbeit: erste ideen fuer biksaetze und VGL !!?, eo
%%%%auch bei VOLLST Isometrie klar?; wohl klar, da meine es ist und gleich wegen Eind (von meiner!!)!!
%%%%---> wie ists bei der andern!?: eindeutigkeiten zusammenbringen!? , eo !?
%etwa
%
%%%%%%%%%%genauere Angaben!!!?
\end{satz}
\begin{proof}
This follows from \cite{mnw},
p.\ 284, \S 2 and Prop.\ 2.2;
condition (iv) in
Definition \ref{amdef} is shown in \cite{hof}, Satz 3.1.
\end{proof}
We close this section by remarking that several properties of the mapping
$\Phi \mapsto \Phi^{(\infty)}$ are studied in a more general situation by Effros and Ruan in \cite{bim} (see p.\ 151 and
Thm.\ 4.2).

\section{The main construction} \label{mainconst}

%Before proving our main result concerning the existence of an amplification in the sense of Definition \ref{amdef}
%and its explicit form,
%
%
We now come to our central result 
concerning the existence of an amplification in the sense of Definition \ref{amdef}
and, which is most important, presenting its explicit form. 
\begin{satz} \label{amsatz}
Let ($\em$, $\en$) be an admissible pair. Then there exists
%$w^*$-$w^*$-stetige isometrische Abbildung
an amplification
$$\Chi: \C\B(\em) \longrightarrow \C\B(\em \tt \en)$$
in the sense of Definition \ref{amdef}. The amplification is explicitly given by
$$\langle \Chi(\Phi) (u), \rho \otimes \tau \rangle =
\langle \Phi (L_\tau (u)), \rho \rangle ,$$
where $\Phi \in \C\B(\em)$, $u \in \em \tt \en$, $\rho \in \em_*$, $\tau \in \en_*$. -- Here, $L_\tau$ denotes
the left slice map associated with $\tau$.
%as introduced by Tomiyama.
%so da"s
%$$\chi(\Phi)(S \otimes T) = \Phi(S) \otimes T$$
%f\"ur alle $\Phi \in \C\B(\em)$, $S \in \em$, $T \in \en$.
\end{satz}
%\begin{bem}
%In the classical framework of $W^*$-algebras, given just a bounded linear operator
%$$\Phi: \B(\H) \longrightarrow \B(\H),$$ one has to assume normality of $\Phi$ to
%insure the existence of an amplification
%$$\Phi \oo \id: \B(\H) \oo \B(\Ka) \longrightarrow \B(\H) \oo \B(\Ka).$$
%This generalzes the well-known result that it is possible to amplify normal completely
%bounded maps on von Neumann algebras
%\end{bem}
\begin{proof}
%We will give an explicit formula for
%Allem voran betonen wir, da"s nach Definition des projektiven Operatorraum-Tensorproduktes gilt:
%$$$$
%
We construct a complete quotient mapping
$$\kappa: \C\B(\em \tt \en)_{*} \longrightarrow \C\B(\em)_{*}$$ in such a way that
$\Chi \dogl \kappa^{*}$ will enjoy the desired properties.
\par
We first note that
%\footnote{The (completely isometric) identification $(\em \oo \en)_{*}
%= \em_{*} {\widehat{\otimes}} \en_{*}$ for
%%arbitrary
%von Neumann algebras $\em$ and $\en$ relies on the properties
%of the projective {\textit{operator space}} tensor product -- whereas the classical theory of Banach space tensor
%products does not provide us with such a formula. This could explain to some extent the different behaviour of these
%two theories with respect to what we stressed in Remark 1.}
%%DAS IRGENDWIE DOCH SCHREIOBEN?, LASSEN??!!
\begin{eqnarray}
\C\B(\em)_{*} \cbgl \em_* {\widehat{\otimes}} \em \label{kan1}
\end{eqnarray}
and, analogously,
\begin{eqnarray}
\C\B(\em \tt \en)_{*} \cbgl (\em \tt \en)_* {\widehat{\otimes}} (\em \tt \en) \label{kan2}
%&=& \em_{*} {\widehat{\otimes}} \en_{*} \tp (\em \tt \en) \label{kan2}
\end{eqnarray}
with completely isometric identifications.
%sagen?: die anderen Identifizierungen verschweigen/stillschweigend vornehmen/machen, aber einmal oder
%beim ersten Mal sagen!? : nicht unbedingt!!
\\
%Weiterhin
We further write
$$W: \C\B(\em \oo \en, \C\B(\en_{*}, \em)) \longrightarrow
\C\B(\en_* \tp (\em \oo \en), \em)$$
for the canonical complete isometry.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\newpage
%In \cite{efru}, Thm.\ 3.2, Effros and Ruan construct a mapping
For $\tau \in \en_{*}$, let us consider
%Tomiyama's
the left slice map
%%%%\footnote{For a detailed discussion, cf.\ \cite{tomiy}, and
%%%%\cite{kraus}, \S 1, p.\ 359 f.}
$$L_{\tau}: \em \oo \en \longrightarrow \em,$$ where
$$L_{\tau}(S \otimes T) = \langle \tau, T \rangle ~S$$ for $S \in \em$, $T \in \en$.
In \cite{ru}, Thm.\ 3.4 (and Cor.\ 3.6, 3.7),
%\cite{efru}, Thm.\ 3.2 and proof, Effros and
Ruan constructs a complete isometric isomorphism (cf.\ also \cite{efru}, Thm.\ 3.2)
%und toll, dass es keiner gemerkt hat, obwohl er-ergebnis und methoden und
%tomiy's slice map klassisch !! sagen!?
$$\theta: \em_* {\widehat{\otimes}} \en_* \longrightarrow (\em \tt \en)_*,$$
where for elementary tensors we have ($S \in \em$, $T \in \en$, $\rho \in \em_*$, $\tau \in \en_*$):
\begin{eqnarray}
\langle \theta(\rho \otimes \tau), S \otimes T \rangle = \langle \rho, S \rangle \langle \tau, T \rangle. \label{ergl}
%= \langle  \rangle. \label{ergl}
\end{eqnarray}
Thus for all $\rho \in \em_*$ and $\tau \in \en_*$, we see that $\theta(\rho \otimes \tau)=\rho \otimes \tau$
on the algebraic tensor product $\em \otimes \en$; but since $\theta(\rho \otimes \tau), \rho \otimes \tau
\in (\em \tt \en)_*$
are normal functionals,
%ist
%we obtain
%schlie"slich
%that
on the whole space $\em \tt \en$ we obtain for all $\rho \in \em_*$, $\tau \in \en_*$:
\begin{eqnarray}
\theta(\rho \otimes \tau)=\rho \otimes \tau. \label{theid}
%$\theta={\mathrm{id}}_{?????!!!!}$
\end{eqnarray}
Hence, in the sequel, we shall not explicitly note $\theta$ when applied on elementary tensors, except for special emphasis.
%nur noch in gewissen F\"allen zur
%Verdeutlichung explizit ausschreiben.
%, ansonsten die Tensoren als Elemente ... lesen: macht Tomiyama aber ja eh, daher weglassen!
%h\"aufig nicht
%So erhalten wir freilich eine vollst\"andige Isometrie
\par
We now obtain a complete isometry:
%from the above identification:
$$\theta^*: \em \tt \en \longrightarrow \C\B(\en_*, \em).$$
For each $\tau \in \en_*$, we define a linear continuous
mapping $$\varphi_{\tau}: \em {\overline{\otimes}} \en \longrightarrow \em$$
through
$$\varphi_{\tau}(u) \dogl \theta^*(u)(\tau)$$
%\in \em
for $u \in \em \tt \en$.
%Sei nun $\tau \in \en_*$. Dann setzen wir f\"ur jedes $u \in \em \tt \en$, we set
%$\varphi_{\tau}(u) \dogl \theta^*(u)(\tau) \in \em$. It is easily seen that $\varphi_{\tau}$ is
%(i)
%Man sieht
%leicht
%unmittelbar
One immediately deduces from the definition that $\varphi_{\tau}$
is normal. Furthermore,
%und
%(ii)
$\varphi_{\tau}$ satisfies
$$\varphi_{\tau} (S \otimes T) = \langle \tau, T \rangle ~S$$
%f\"ur alle
for all $S \in \em$, $T \in \en$. -- For if $\rho \in \em_*$, we have:
\begin{eqnarray*}
\langle \varphi_{\tau}(S \otimes T), \rho \rangle &=&
\langle \theta^*(S \otimes T)(\tau), \rho \rangle \\
&=& \langle \theta^*(S \otimes T), \rho \otimes \tau \rangle \\
&=& \langle \theta(\rho \otimes \tau), S \otimes T \rangle \\
&\stackrel{(\ref{ergl})}{=}& \langle \rho, S \rangle \langle \tau, T \rangle.
\end{eqnarray*}
%erf\"ullt.
%Es folgt nun aus der Eindeutigkeit von Tomiyamas linker slice map, da"s
Hence, $\varphi_{\tau}$ coincides with the left slice map $L_\tau$ on all elementary tensors in $\em \tt \en$, and since
both mappings are normal,
%\footnote{Tomiyamas slice map $L_{\tau}$ ist bekanntlich $\sigma$-schwach-stetig,
%also normal.},
we finally see that
$\varphi_{\tau}=L_{\tau}$. Thus we obtain $$\theta^*(u)(\tau) = L_{\tau} (u)$$ for all $u \in \em \tt \en$, $\tau \in \en_*$.
%\par
%To see (i), let $(u_{\alpha}) \subseteq \Ball(\em \tt \en)$ be a net converging $w^*$ to 0. Then for
%$\rho \in \em_*$:
%\begin{align}
We note that, of course,
$$\theta^* \in \C\B(\em \tt \en, \C\B(\en_*, \em)).$$
%We have $$\theta \in \C\B(\B(\H) \oo \B(\Ka), \C\B(\Te(\Ka), \B(\H))$$ which means
%\par
Hence, by definition of $W$, we have:
%demzufolge:
%also:
%nun
$$W(\theta^*) \in \C\B(\en_* \tp (\em \oo \en), \em).$$
%We finally put
Thanks to the functorial property of the projective operator space tensor product, we now get
%endlich
%unsere ersehnte
a completely bounded mapping
%weitere Eigenscvhaften!!???, auch fuer slice map?, eo!?
$$\id_{\em_*} \tp W(\theta^*):
\em_* \tp \en_* \tp (\em \oo \en) \longrightarrow
\em_* {\widehat{\otimes}} \em.$$
What remains to be done, is the passage from $(\em \tt \en)_* \tp (\em \tt \en)$ to
$\em_* \tp \en_* \tp (\em \oo \en)$. This is accomplished by the complete quotient mapping
%(beachte hierzu, da"s das
%projektive Operatorraum-Tensorprodukt vollst\"andige Quotientenabbildungen erh\"alt)
$$\theta^{-1} \tp {\mathrm{id}}_{\em \tt \en}: (\em \tt \en)_* \tp (\em \tt \en)
\longrightarrow \em_* \tp \en_* \tp (\em \oo \en).$$
%Schalten wir nun letztere Abbildung vor $\id_{\em_*} \tp W(\theta^*)$, so erhalten wir mit
Now, defining
$$\kappa \dogl (\id_{\em_*} \tp W(\theta^*)) \circ (\theta^{-1} \tp {\mathrm{id}}_{\em \tt \en}): (\em \tt \en)_* \tp (\em \tt \en)
\longrightarrow \em_* {\widehat{\otimes}} \em,$$
we obtain a completely bounded mapping which has the desired properties, as we shall now prove.
\par
In view of (\ref{kan1}) and (\ref{kan2}), we see that
%mit $\kappa$
we have indeed constructed
%eine vollst\"andig beschr\"ankte Abbildung
a completely bounded mapping
$$\kappa: \C\B(\em \tt \en)_{*} \longrightarrow \C\B(\em)_{*},$$
%konstruiert haben, und es bleiben lediglich die -- nunmehr nahezu offensichtlichen -- Eigenschaften der
%(freilich auch)
%vollst\"andig beschr\"ankten Abbildung
and it remains to verify the properties of the mapping
$$\Chi \dogl \kappa^*: \C\B(\em) \longrightarrow \C\B(\em \tt \en).$$
%zu verifizieren.
\par
%Wir sehen nun,
First, we remark the following:
%Aufgrund der Eigenschaften der vollst\"andigen Isometrie
\par
$(*) ~{}$
Since the mapping 
$\theta: \em_* \tp \en_* \longrightarrow (\em \tt \en)_*$ is a complete isometry,
%sagen: unter vollst isometrie verstehen wir surj dabei!!!!
%ist, sieht man sehr leicht, da"s
it is readily seen that
$\theta(\em_* \otp \en_*) \otp (\em \tt \en)$ is norm dense in $(\em \tt \en)_* \tp (\em \tt \en)$.
% liegt -- freilich
%unter mehrfacher Nutzung der Dichtheit
%von $$
%des algebraischen Tensorproduktes im projektiven Operatorraum-Tensorprodukt.
Every element of the space $\C\B(\em \tt \en) \cbgl ((\em \tt \en)_* \tp (\em \tt \en))^*$ is thus completely determined by
its values on
%ist also bereits durch seine Werte auf
$\theta(\em_* \otp \en_*) \otp (\em \tt \en)$.
\par
%Zuvor aber betonen wir, da"s nach
%Wir betonen zun\"achst, da"s nach
%Definition
%Konstruktion
By construction, we see that for all $\Phi \in \C\B(\em)$, $u \in \em \tt \en$, $\rho \in \em_*$ and $\tau \in \en_*$:
%%oder: (wobei ...), eo
\begin{eqnarray*}
\langle \Chi(\Phi), \theta(\rho \otimes \tau) \otimes u \rangle &=& \langle \Phi, \kappa(\theta(\rho \otimes \tau) \otimes u)
\rangle \\
&=& \langle \Phi, \rho \otimes W(\theta^*)(\tau \otimes u) \rangle \\
&=& \langle \Phi, \rho \otimes \theta^*(u)(\tau) \rangle \\
&=& \langle \Phi, \rho \otimes L_{\tau}(u) \rangle.
\end{eqnarray*}
%f\"ur alle
%so da"s wir insgesamt festhalten:
Hence we have:
\begin{eqnarray*}
\langle \Chi(\Phi), \theta(\rho \otimes \tau) \otimes u \rangle =
\langle \Phi, \rho \otimes L_{\tau}(u) \rangle,
\end{eqnarray*}
and following $(*)$, by this $\Chi(\Phi)$ is completely determined. Taking into account (\ref{theid}), we thus have
that for all $\rho \in \em_*$, $\tau \in \en_*$, $u \in \em \tt \en$:
\begin{eqnarray}
\langle \Chi(\Phi), \rho \otimes \tau \otimes u \rangle =
\langle \Phi, \rho \otimes L_{\tau}(u) \rangle, \label{chi}
\end{eqnarray}
which determines $\Chi(\Phi)$ completely.
%besser hier theta weglassen als oben bei bem (*)
\par
Now,
let us verify
the condition (AAC) and
%conditions
properties (i)--(iv).
%isometrie: slice maps!?, eo
%Hierzu
\par
(AAC) Fix
$\Phi \in \C\B(\em)$, $S \in \em$, $T \in \en$,
$\rho \in \em_*$ and $\tau \in \en_*$.
We obtain:
%das Gew\"unschte, n\"amlich:
\begin{eqnarray*}
\langle \rho \otimes \tau, \Chi (\Phi) (S \otimes T) \rangle &=&
\langle \Chi(\Phi), \rho \otimes \tau \otimes (S \otimes T) \rangle \\
&\stackrel{(\ref{chi})}{=}& \langle \Phi, \rho \otimes L_{\tau}(S \otimes T) \rangle \\
%\langle \kappa((S \otimes T) \otimes \rho \otimes \tau), \Phi \rangle
%&=& \langle \langle \tau, T \rangle S \otimes \rho, \Phi \rangle \\
&=& \langle \tau, T \rangle \langle \Phi, \rho \otimes S \rangle \\
&=& \langle \rho, \Phi(S) \rangle \langle \tau, T \rangle \\
&=& \langle \rho \otimes \tau, \Phi(S) \otimes T \rangle.
\end{eqnarray*}
%und damit das Gew\"unschte:
Hence, for all $\Phi \in \C\B(\em)$, $S \in \em$ and $T \in \en$, we have:
$$\Chi(\Phi)(S \otimes T) = \Phi(S) \otimes T,$$ as desired.
%Offensichtlich
%%%%%%%%%%%%%%%%%%%Weiterhin ist $\chi$ als adjungierte Abbildung nat\"urlich $w^*$-$w^*$-stetig.
\par
(i) To prove that $\Chi$ is a complete isometry,
we show that $\kappa=\Chi_*$ is a complete quotient mapping. But this follows from the fact that
%(da das projektive Operatorraum-Tensorprodukt
%Quotientenabbildungen erh\"alt),
$W(\theta^*)$ is a complete quotient mapping. (For if this is established, then the mapping
$\id_{\em_*} \tp W(\theta^*) \in \C\B(\em_* \tp \en_* \tp (\em \tt \en), \em_* \tp \em)$
%auf der Grundstufe
is a complete quotient map,
%??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
and of course this is also the case for
%%die vollst\"andige Quotientenabbildung
%(beachte, da"s dies freilich f\"ur
$\theta^{-1} \tp \id_{\em \tt \en} \in \C\B((\em \tt \en)_* \tp (\em \tt \en), \em_* \tp \en_* \tp (\em \tt \en))$.)
%%%denn zu letzterer Abb schon gesagt, dass es eine vollst Q-Abb!!
%sogar vollste q-abb!?, eo; und bezug zu oben, weitere Eigen sonst eeeooo
%gen\"ugt es freilich zu zeigen, da"s
%$W(\theta^*)$ -- und damit $\kappa$ -- Quotientenabbildungen sind.
This in turn is easily seen. Fix $n \in \N$. Let
$S=[S_{i,j}]_{i, j} \in M_n(\em)$, $\|S\| < 1$. We are looking for an element $\varphi \in M_n(\en_* \tp (\em \tt \en))$, such that
$\| \varphi \| < 1$ and $W(\theta^*)^{(n)}(\varphi)=S$, where $W(\theta^*)^{(n)}$ denotes the $n$th amplification of the mapping
$W(\theta^*)$.
\par
Let us choose
$\tau \in
\en_*$ with $\|\tau\|=1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%geht nicht wenn Raum = Nullraum: nicht ueberlegen!?, eo !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ???????????
Then there exists $T \in \en$, $\|T\|=1$, such that $\langle \tau, T \rangle = 1$.
Now
%consider
$\varphi \dogl [\tau \otimes (S_{i, j} \otimes T)]_{i, j} \in M_n(\en_* \tp (\em \tt \en))$ satisfies all our requirements.
For we have:
%Denn zun\"achst erkennen wir
%(unter Nut
%Kreuznorm eh nichjt richtig definiert!: oder!?; vgl lexikon und witt vs bl-pau!?, eo
\begin{eqnarray*}
\| \varphi \|_{M_n(\en_* \tp (\em \tt \en))} &=&
\| \tau \| ~\| [S_{i, j} \otimes T]_{i, j} \|_{M_n(\em \tt \en)} \\
&=& \| \tau \| ~\| [S_{i, j} \otimes T]_{i, j} \|_{M_n(\em \ti \en)} \\
&=& \| \tau \| ~\| [S_{i, j}]_{i, j} \|_{M_n(\em)} ~\| T \| \\
&<& 1.
\end{eqnarray*}
%Wir erhalten demnach
Furthermore, we see that:
\begin{eqnarray*}
W(\theta^*)^{(n)}(\varphi) &=& W(\theta^*)^{(n)}([\tau \otimes (S_{i, j} \otimes T)]_{i, j}) \\
&=& [W(\theta^*)(\tau \otimes (S_{i, j} \otimes T))]_{i, j} \\
&=& [\theta^*(S_{i, j} \otimes T)(\tau)]_{i, j} \\
&=& [L_{\tau} (S_{i, j} \otimes T)]_{i, j} \\
&=& [\langle \tau, T \rangle S_{i, j}]_{i, j} \\
&=& S,
%[S_{i, j}]_{i, j}
%W(\theta^*)(\tau \otimes (S \otimes T)) = \theta^*(S \otimes T)(\tau) =
%L_{\tau} (S \otimes T) = \langle \tau, T \rangle S
%&=& S,
%wobei
%$\| \tau \otimes (S \otimes T) \| < 1$.
\end{eqnarray*}
which establishes the claim.
\par
(ii) We show that $\Chi$ is an algebra homomorphism.
-- To this end, let $\Phi, \Psi \in \C\B(\em)$. We have to show that
$$\Chi(\Phi \Psi) = \Chi(\Phi) \Chi(\Psi),$$
%%wie ueber normalitaet vorgehen!?, eo
as elements in $$\C\B(\em \tt \en) \cbgl (\em_* \tp \en_* \tp (\em \tt \en))^*.$$
Fix $T \in \em \tt \en$, $\rho \in \em_*$ and $\tau \in \en_*$. It suffices to show that
$$\langle \Chi(\Phi \Psi), \rho \otimes \tau \otimes T \rangle =
\langle \Chi(\Phi) \Chi(\Psi), \rho \otimes \tau \otimes T \rangle.$$
On the left side, we obtain:
\begin{eqnarray*}
\langle \Chi(\Phi \Psi), \rho \otimes \tau \otimes T \rangle &\stackrel{(\ref{chi})}{=}&
%\langle \Phi \Psi, \kappa(T \otimes \rho \otimes \tau) \rangle \\
\langle \Phi \Psi, \rho \otimes L_{\tau}(T) \rangle \\
%passt dieses ruebershiften zu tomiyamas definitoirischem slmap-shiften!? Bezuege!?
&=& \langle \Phi (\Psi(L_{\tau}(T))), \rho \rangle \\
&=& \langle \Phi, \rho \otimes \Psi(L_{\tau}(T)) \rangle.
\end{eqnarray*}
On the right, we have:
%in umgekehrt-analoger Weise
\begin{eqnarray*}
\langle \Chi(\Phi) \Chi(\Psi), \rho \otimes \tau \otimes T \rangle &=&
\langle \Chi(\Phi)[\Chi(\Psi)(T)], \rho \otimes \tau \rangle \\
&=& \langle \Chi(\Phi), \rho \otimes \tau \otimes \Chi(\Psi)(T) \rangle \\
%&=& \langle \Phi, \kappa[\Chi(\Psi)(T) \otimes \rho \otimes \tau] \rangle \\
&\stackrel{(\ref{chi})}{=}& \langle \Phi, \rho \otimes L_{\tau}(\Chi(\Psi)(T)) \rangle.
\end{eqnarray*}
Thus we have to show that
$$L_{\tau}(\Chi(\Psi)(T)) = \Psi(L_{\tau}(T)),$$ as elements in $\em$.
%schoene formel!!: mit diagramm in bem hervorheben!?, eo
To this end, let $\rho \in \em_*$.
We then have:
\begin{eqnarray*}
\langle L_{\tau}(\Chi(\Psi)(T)), \rho \rangle &=& \langle \Chi(\Psi)(T), \rho \otimes \tau \rangle \\
&=& \langle \Chi(\Psi), \rho \otimes \tau \otimes T \rangle \\
&\stackrel{(\ref{chi})}{=}& \langle \Psi, \rho \otimes L_{\tau}(T) \rangle \\
&=& \langle \Psi(L_{\tau}(T)), \rho \rangle,
\end{eqnarray*}
which yields the desired equality.
%sagen: ohne fortsetzsatz uund/oder dergl !?, eo
\par
(iii) Being an adjoint mapping, $\Chi$ is clearly $w^*$-$w^*$-continuous.
%As an adj mapping, ...
\par
(iv) Fix $\Phi \in \C\B^{\sigma}(\em)$. We wish to prove that $\Chi(\Phi) \in \C\B^{\sigma}(\em \tt \en)$.
So let $(T_{\alpha}) \subseteq \Ball(\em \tt \en)$ be a net such that $T_{\alpha} \stackrel{w^*}{\longrightarrow} 0$.
We claim that $\Chi(\Phi)(T_{\alpha}) \longrightarrow 0$ ($\sigma(\em \tt \en, \em_* \tp \en_*)$).
It suffices to show that we have for arbitrary $\rho \in \em_*$ and $\tau \in \en_*$:
$$\langle \Chi(\Phi)(T_{\alpha}), \rho \otimes \tau \rangle \longrightarrow 0.$$
%Nun ist aber
We obtain for all indices $\alpha$:
\begin{eqnarray*}
\langle \Chi(\Phi)(T_{\alpha}), \rho \otimes \tau \rangle &=& \langle \Chi(\Phi), \rho \otimes \tau \otimes T_{\alpha} \rangle \\
&\stackrel{(\ref{chi})}{=}& \langle \Phi, \rho \otimes L_{\tau}(T_{\alpha}) \rangle \\
&=& \langle \Phi(L_{\tau}(T_{\alpha})), \rho \rangle.
\end{eqnarray*}
%Due to the normality of
Since $\Phi$ is normal, it thus remains to show that
$L_{\tau}(T_{\alpha})
\stackrel{w^*}{\longrightarrow} 0$; but this follows in turn from the normality of the slice map
%bekanntlich ist Tomiyamas slice map $L_{\tau}$ eine $\sigma$-schwach-stetige Abbildung, also
$L_{\tau}$.
%noch sagen, woher (zitieren!), dass L_tau sigma-schw.stetig!??? Kraus oder tom-vorl eeoo
%noch sagen vor isometrie-beweis: Norm chi = 1 klar. :??
\end{proof}
%\begin{kor}
%Let $\Ar$ be a Banach space, and let $\H$ and $\Ka$ be Hilbert spaces.
%Let further $\Psi: \Ar^* \longrightarrow \C\B(\B(\H))$ be a bounded linear mapping, which in addition is
%$w^*$-$w^*$-continuous [isometric].
%Then there exists a
%Amplifikation
%bounded linear mapping $\widetilde{\Psi}: \Ar^* \longrightarrow
%\C\B(\B(\H \otimes_2 \Ka))$, which is
%vollst\"andig beschr\"ankt und
%$w^*$-$w^*$-continuous [isometric] and is an amplification of $\Psi$ in the following sense: For
%If...
%$m \in \Ar^*$, $S \in \B(\H)$, and $T \in \B(\Ka)$, one has:
%$$(\widetilde{\Psi}(m))(S \otimes T) = \Psi(m)(S) \otimes T.$$
%If in addition $\Ar^*$ is a Banach algebra and $\Psi$ is an algebra homomorphism, then there exists
%ist auch
%a mapping $\widetilde{\Psi}: \Ar^* \longrightarrow
%\C\B(\B(\H \otimes_2 \Ka))$ with the above properties, which is also an algebra homomorphism.
%\end{kor}
%\begin{proof}
%Apply Theorem \ref{amsatz}, putting $\widetilde{\Psi} \dogl \Chi \circ \Gamma$.
%\end{proof}
%\begin{prop}
%Let $\em$ and $\en$ be arbitrary von Neumann algebras.
%Every completely bounded map
%$$\Phi \in \C\B(\em)$$ admits an amplification to a map
%$$\Phi {\overline{\otimes}} \id \in \C\B(\em {\overline{\otimes}} \en).$$
%%oben id_{en} schreiben/nehmen!!????
%The corresponding assignment $$\chi: \C\B(\em) \longrightarrow \C\B(\em {\overline{\otimes}} \en)$$
%enjoys the above mentioned properties of normality and isometry.
%\end{prop}
%\begin{proof}
%We use in
%%the same
%an analogous way as we did in the proof of Theorem 1 Tomiyama's left slice map
%$$L_{f}: \em {\overline{\otimes}} \en \longrightarrow \em,$$
%where $f \in \en_{*}$, and the (completely isometric)
%identification
%$$\C\B(\em {\overline{\otimes}} \en)_{*} = (\em {\overline{\otimes}} \en)
%{\widehat{\otimes}} \em_{*} {\widehat{\otimes}} \en_{*}.$$
%\end{proof}

\section{Discussion of the amplification mapping} \label{disc}
%Properties a variants of the ...
In the sequel, for a mapping $\Phi \in \C\B(\em)$, we shall consider amplifications with respect to different
von Neumann algebras or dual operator spaces $\en$.
We will show that all of these amplifications are compatible with each other in a very natural way.
%Insbesondere werden wir ein Resultat herleiten, welches die nat\"urliche Kompatibilit\"at
%all dieser Amplifikationen sichert. Hierzu
%ist
For this purpose, we introduce the following
\begin{bez} \label{chin}
If ($\em$, $\en$) is an admissible pair, we will denote by $\Chi_{\en}$
the amplification with respect to $\en$ as constructed
in Theorem \ref{amsatz}.
\end{bez}
Taking the restriction of the mapping $\Chi_{\en}: \C\B(\em) \longrightarrow \C\B(\em \tt \en)$ to the subalgebra
$\C\B^{\sigma}(\em)$, by condition (iv) in Definition \ref{amdef}, we obtain a mapping
$\Chi_0: \C\B^{\sigma}(\em) \longrightarrow \C\B^{\sigma}(\em \tt \en)$, which of course satisfies the algebraic condition
(AAC) on $\C\B^{\sigma}(\em)$.
%Bezeichnen wir mit $H$ die von de Canni\`ere und Haagerup in \cite{haa}, Lemma 1.5 b konstruierte
%Abbildung,
%Amplifikation,
But this is the case as well for the amplification
$$H: \C\B^{\sigma}(\em) \longrightarrow \C\B^{\sigma}(\em \tt \en)$$
%constructed
considered, for von Neumann algebras $\em$ and $\en$, by
de Canni\`ere and Haagerup in \cite{haa}, Lemma 1.5 (b).
%Abbildung,
%wollen
%sie
Note that $H$
%wie es jede abb waere mit selbem ur- und bildbereich!
is completely determined by
(AAC), since the image of $H$ consists of normal mappings.
%(cf.\ auch den Beweis
%von Pro\-po\-si\-tion \ref{eindam}).
Hence we deduce that $\Chi_0=H$, thus showing that $\Chi_{\en}$ is indeed a ($w^*$-$w^*$-continuous) extension of $H$
to $\C\B(\em)$, where $\em$ and $\en$ are von Neumann algebras (cf.\ also \cite{kraus}, p.\ 123, for a
%brief
discussion of the mapping
$H$ in a more general setting). 
\begin{bem} 
Theorem \ref{amsatz} 
%, it 
%clearly 
%follows 
implies in particular (cf.\ (iv) in Definition \ref{amdef}) that 
whenever $(\em , \en)$ is an admissible pair, then for every $\Phi \in \C\B^\sigma (\em)$, 
the mapping $\Chi_\en (\Phi)$ is the unique normal map in $\C\B^\sigma (\em \tt \en)$ which satisfies (AAC). 
This entails that the last assumption made in \cite{krtrans}, Prop.\ 1.19 is always fulfilled (and even true in greater 
generality than actually needed there), which besides slightly simplifies the proof of \cite{krtrans}, Thm.\ 2.2. 
\end{bem} 
%Man kann also in dem Ergebnis von de Canni\`ere und Haagerup auf die Normalit\"at der Operatoren $\Phi$
%verzichten.
%We
Let us now briefly return to the special case where $\em=\B(\H)$ and $\en=\B(\Ka)$.
We already
%stressed
noted in Theorem \ref{wmnek} that the mapping
$\Phi \mapsto \Phi^{(\infty)}$, where $\Phi \in \C\B(\B(\H))$, is an amplification
in the sense of Definition \ref{amdef}. Since $\B(\H)$ trivially
%of course
is an injective factor, we conclude by Proposition \ref{eindam} that
\begin{eqnarray}
\Chi_{\B(\Ka)} (\Phi) = \Phi^{(\infty)} \label{glinf}
\end{eqnarray}
for all $\Phi \in \C\B(\B(\H))$.
%Die obige Konstruktion
Thus, restricted to that special case,
%in this sp case, ...
Theorem \ref{amsatz} provides
%einen
%neuen Zugang //approach!?, eo
a new construction of the mapping $\Phi \mapsto \Phi^{(\infty)}$ and for the first time shows the intimate relation to
Tomiyama's slice
maps.
%%%%%%%auch sl maps bei stratila!?, eo: aber laesst sich wohl nicht durchhalten bei witt-fortsetzung!?, eo eeoo
\par
We finally stress that
%aus der Konstruktion von $\Chi_{\B(\Ka)}$ unmittelbar
our construction of $\Chi_{\B(\Ka)} (\Phi)$ is coordinate free -- in contrast to the
%derselben
definition of $\Phi^{(\infty)}$ in \cite{mnw}:
%auch effkis immer oder wann erwaehnen, eo!!?????????????????????????????????????????????????
%koordinatenfrei --
since
%we are not using an
our approach does not rely on an
explicit representation of operators in $\B(\H) \tt \B(\Ka)$
as infinite matrices, we do not have to choose a basis in the Hilbert space $\Ka$, whereas in \cite{mnw}, the latter is
represented as $\ell_2(I)$ for some suitable $I$. The equality
$$\Chi_{\B(\Ka)} (\Phi) = \Phi^{(\infty)} \quad (\Phi \in \C\B(\B(\H))),$$
obtained above, now immediately implies that
%liefert
%Koordinatenfreiheit
the definition of the mapping $\Phi^{(\infty)}$ in \cite{mnw} does not depend on the particular choice of a basis. (This fact
%can of course
of course can also be obtained intrinsically from the construction of $\Phi^{(\infty)}$, but it comes for free in our approach.)
%but we prefer...
%but with more effort...
%Annahme des obigen \glqq Geschenks\grqq.)
%auch sagen
%darstellungsfrei?: so sagen? - Darstellungsfreiheit von CB_R(B(H)) kriege ich da irgendwoie auch was dazu?
%und oder mit wmn eo -sachen zusammennehmen!? eeeooo
\par
We now come to the result announced above
%concerning
which establishes the compatibility of our amplification with respect to different
von Neumann algebras or dual operator spaces $\en$, respectively.
\begin{satz} \label{kompa}
%Let $\en_0$ and $\en$ be von Neumann algebras with $\en_0 \subseteq \en$.
Let ($\em$, $\en$) be an admissible pair. Let further $\en_0 \subseteq \en$ be either a von Neumann subalgebra
%of $\en$
or a dual operator subspace of $\en$, respectively (according to case (i) or (ii) of Definition \ref{pair}).
%Let $\en$ be a von Neumann algebra, and $\en_0 \subseteq \en$ be a von Neumann subalgebra.
Then, for all
$\Phi \in \C\B(\em)$, we have:
$$\Chi_{\en}(\Phi) |_{\em \tt \en_0} = \Chi_{\en_0}(\Phi).$$
\end{satz}
%Zum Beweis ben\"otigen wir zun\"achst ein kleines
%\begin{lemma}
%\end{lemma}
\begin{proof}
Denote by $\iota: \em \tt \en_0 \hookrightarrow \em \tt \en$ the canonical embedding, and write
$q: \en_* \twoheadrightarrow (\en_0)_*$
for the
%kanonische Quotientenabbildung
restriction map. Obviously, the pre-adjoint mapping $\iota_*: \em_* \tp \en_* \twoheadrightarrow
\em_* \tp (\en_0)_*$ satisfies
$\iota_* = \id_{\em_*} \tp q$.
\par
We
%will show
shall prove that the following diagram commutes:
%rot!
%\begin{center}
$$\xymatrix{
%\begin{center}
\em \tt \en \ar[r]_{\Chi_{\en}(\Phi)} & \em \tt \en \\
\em \tt \en_0 \ar@{^{(}->}[u]_{\iota} \ar[r]^{\Chi_{\en_0}(\Phi)} & \em \tt \en_0 \ar@{^{(}->}[u]^{\iota}
%\end{center}
}
$$
%\end{center}
This yields the desired equality (and shows at the same time that
$\Chi_{\en}(\Phi) |_{\em \tt \en_0}$
indeed
leaves the space $\em \tt \en_0$ invariant).
%%%%%%%%%LETZTERES EIGLICH UNNOETIG, DENN nach equality wegen rechter seite derselben eh klar!!
\par
%Mit anderen Worten:
To this end, fix $u \in \em \tt \en_0$. We have to show that
$$\Chi_{\en}(\Phi)(\iota(u)) = \iota(\Chi_{\en_0}(\Phi)(u)).$$
Let $\rho \in \em_*$ and $\tau \in \en_*$. It is sufficient to verify that
$$\langle \Chi_{\en}(\Phi)(\iota(u)), \rho \otp \tau \rangle =
\langle \iota(\Chi_{\en_0}(\Phi)(u)), \rho \otp \tau \rangle.$$
For the left-hand expression, we get:
%The left-......
$$\langle \Chi_{\en}(\Phi)(\iota(u)), \rho \otp \tau \rangle =
\langle \Chi_{\en}(\Phi), \rho \otp \tau \otimes
\iota(u) \rangle = \langle \Phi, \rho \otp L_{\tau}(\iota(u)) \rangle.$$
On the right-hand side, we find:
\begin{eqnarray*}
\langle \iota(\Chi_{\en_0}(\Phi)(u)), \rho \otp \tau \rangle &=&
\langle \Chi_{\en_0}(\Phi)(u), \rho \otp q(\tau) \rangle \\ &=&
\langle \Chi_{\en_0}(\Phi), \rho \otp q(\tau) \otp u \rangle \\ &=&
\langle \Phi, \rho \otimes L_{q(\tau)}(u) \rangle.
\end{eqnarray*}
It thus remains to be shown that
$$L_{\tau}(\iota(u)) = L_{q(\tau)}(u).$$
%Let
%. We obtain:
But for $\varphi \in \em_*$, we obtain:
\begin{eqnarray*}
\langle L_{\tau}(\iota(u)), \varphi \rangle &=&
\langle \iota(u), \varphi \otp \tau \rangle \\ &=&
\langle u, \varphi \otp q(\tau) \rangle \\ &=&
\langle L_{q(\tau)}(u), \varphi \rangle,
%%%%%%%toll: einfache, elegante, straightforward-linalg-Beweise
%%%%%%%recht tiefer und jedenfalls schoener saetze: nicht mit ganzer
%%%%%%%theorie wie witt! und kuerzer und durchsichtiger:
%%%%%%%beweismoegl von von ihm benutzten saetzen!?, eo; vgl wmn-Artikel!
\end{eqnarray*}
which yields the claim.
\end{proof}
%
%
%Wir schlie"sen unsere Diskussion mit einem vorl\"aufigen R\'esum\'e, das wir in einen hoffnungstr\"achtigen
%Ausblick m\"unden lassen.
%\par
%We
%shall
%will now discuss another very natural
%approach
%strategy to attack the
%%and show that it entails
%finally  ... precisely
%amplification problem, which yields, as we shall prove,
We shall now present another very natural, though highly non-constructive,
approach to the amplification problem, and discuss its
%the
relation to
%the
our concept.
%developed above.
%-- Namely, let
-- Let ($\em$, $\en$) be an admissible pair, with $\em \subseteq \B(\H)$ and $\en \subseteq \B(\K)$.
An element $\Phi \in \C\B(\em)$ may
%now
be considered as a completely bounded mapping from $\em$ with
values in $\B(\H)$. Hence we can assign to $\Phi$ a
%n arbitrary
Wittstock-Hahn-Banach extension $\widetilde{\Phi} \in \C\B(\B(\H))$.
Then the mapping
$$\widetilde{\Phi}^{(\infty)} \mid_{\em \tt \en}: \em \tt \en \longrightarrow \B(\H) \tt \B(\K)$$
is completely bounded and of course satisfies the algebraic amplification condition (AAC).
As we shall now see, this (non-constructive) abstract procedure yields indeed a mapping
which takes
%taking
values in $\em \tt \en$ and
%is independent of
does not depend on the choice of the Wittstock-Hahn-Banach extension --
%given
%Namely, a first attempt to amplify ...
namely, $\widetilde{\Phi}^{(\infty)} \mid_{\em \tt \en}$ is nothing but $\Chi_\en (\Phi)$. This will
be proved in a similar fashion as Theorem \ref{kompa}.
%above.
\begin{satz} \label{ichuwitt}
Let ($\em$, $\en$) be an admissible pair, where $\em \subseteq \B(\H)$ and $\en \subseteq \B(\K)$.
Let further $\Phi \in \C\B(\em)$. Then for an arbitrary
Wittstock-Hahn-Banach extension $\widetilde{\Phi} \in \C\B(\B(\H))$ obtained as above, we have:
$$\widetilde{\Phi}^{(\infty)} \mid_{\em \tt \en} = \Chi_\en (\Phi).$$
\end{satz}
\begin{proof}
%Denote by
We write $\iota: \em \tt \en \hookrightarrow \B(\H) \tt \B(\K)$ for the canonical embedding. It is sufficient to
show that the following diagram commutes:
$$\xymatrix{
%\begin{center}
\B(\H) \tt \B(\K) \ar[r]_{\widetilde{\Phi}^{(\infty)}} & \B(\H) \tt \B(\K) \\
\em \tt \en \ar@{^{(}->}[u]_{\iota} \ar[r]^{\Chi_{\en}(\Phi)} & \em \tt \en \ar@{^{(}->}[u]^{\iota}
%\end{center}
}
$$
Fix $u \in \em \tt \en$, $\rho \in \B(\H)_*$, $\tau \in \B(\K)_*$.
We have only to show that
$$\langle \iota [ \Chi_\en (\Phi) (u)  ], \rho \otimes \tau \rangle =
\langle \widetilde{\Phi}^{(\infty)} (\iota (u)), \rho \otimes \tau \rangle ,$$
which by equation (\ref{glinf}) is equivalent to
\begin{eqnarray}
\langle \iota [ \Chi_\en (\Phi) (u)  ], \rho \otimes \tau \rangle =
\langle \Chi_{\B(\K)} (\widetilde{\Phi})  (\iota (u)), \rho \otimes \tau \rangle . \label{anfan}
\end{eqnarray}
On the left-hand side, we obtain:
\begin{eqnarray*}
\langle \iota [ \Chi_\en (\Phi) (u)  ], \rho \otimes \tau \rangle &=&
\langle \Chi_\en (\Phi) (u) , \rho \mid_\em \otimes~ \tau \mid_\en \rangle \\
&=& \langle \Phi, \rho \mid_\em \otimes~ L_{\tau \mid_\en} (u)  \rangle .
\end{eqnarray*}
Now, writing $\iota' : \em \hookrightarrow \B(\H)$ for the canonical embedding, we note that
\begin{eqnarray*}
\langle L_\tau (\iota (u)), \rho \rangle &=& \langle \iota (u), \rho \otimes \tau \rangle \\
&=& \langle u, \rho \mid_\em \otimes~ \tau \mid_\en \rangle \\
&=& \langle L_{\tau \mid_{\en}} (u), \rho \mid_\em \rangle \\
&=& \langle \iota' (L_{\tau \mid_{\en}} (u)), \rho \rangle ,
\end{eqnarray*}
whence we have:
$$L_\tau (\iota (u)) = \iota' (L_{\tau \mid_{\en}} (u)).$$
Thus, we see that
\begin{eqnarray}
\widetilde{\Phi} [L_\tau (\iota (u))] = \widetilde{\Phi} [\iota' (L_{\tau \mid_{\en}} (u))]
= \iota' [\Phi (L_{\tau \mid_{\en}} (u))] . \label{letzt}
\end{eqnarray}
Now we find for the right-hand side term of equation (\ref{anfan}):
\begin{eqnarray*}
\langle \Chi_{\B(\K)} (\widetilde{\Phi})  (\iota (u)), \rho \otimes \tau \rangle &=&
\langle \widetilde{\Phi} , \rho \otimes L_\tau (\iota (u))  \rangle \\
&=& \langle \widetilde{\Phi} [L_\tau (\iota (u))] , \rho \rangle \\
&\stackrel{(\ref{letzt})}{=}& \langle \iota' [\Phi (L_{\tau \mid_{\en}} (u))] , \rho \rangle \\
&=& \langle \Phi (L_{\tau \mid_{\en}} (u)) , \rho \mid_{\em}   \rangle \\
&=& \langle \Phi, \rho \mid_\em \otimes~ L_{\tau \mid_\en} (u)  \rangle ,
\end{eqnarray*}
whence we deduce the desired equality.
\end{proof}
\begin{bem}
If $\em \subseteq \B(\H)$, $\en \subseteq \B(\K)$ are von Neumann algebras and $\Phi \in \C\P(\em)$ is completely positive, then,
using an arbitrary Arveson extension of $\Phi$ to a completely positive map $\widetilde{\Phi} \in \C\P(\B(\H))$, we obtain by
the same reasoning as above that $\widetilde{\Phi}^{(\infty)} \mid_{\em \tt \en} = \Chi_\en (\Phi)$. But it is
easy to see from the definition of the mapping $\widetilde{\Phi} \mapsto \widetilde{\Phi}^{(\infty)}$
(cf.\ \cite{mnw}, p.\ 284, \S 2)
that $\widetilde{\Phi}^{(\infty)}$ still is completely positive, whence $\Chi_\en (\Phi)$ also is.
%Thus we see
This shows that our amplification $\Chi_\en$ preserves complete positivity.
\end{bem}

\section{An algebraic characterization of 
%Towards the tensor product of completely bounded maps
normality} \label{tow}
In the following, let still ($\em$, $\en$) be an admissible pair.
Using Tomiyama's left slice map, for every completely bounded mapping
$\Phi \in \C\B(\em)$,
we have constructed an amplification
%Wir haben mit Hilfe von Tomiyamas linker slice map eine Amplifikation
$\Chi_{\en}(\Phi) =: \Phi \otp \id_{\en} \in \C\B(\em \tt \en)$.
% bzgl.\ einer
%von Neumann-Algebra $\en$
%f\"ur
%beliebige vollst\"andig beschr\"ankte Abbildungen $\Phi \in \C\B(\em)$ konstruiert.
%Wir schreiben f\"ur diese\footnote{Genauer freilich f\"ur alle so konstruierten; man beachte
%die nicht notwendige Eindeutigkeit.} Amplifikation \glqq in der rechten Variablen\grqq\ $\Chi_{\en, r}$.
%Similarly,
%Vollkommen
In an analogous fashion, replacing in the above construction Tomiyama's left by the appropriate right slice
maps,
we obtain -- {\textit{mutatis mutandis}} -- an amplification of completely bounded mappings
%k\"onnen wir vollst\"andig beschr\"ankte Abbildungen
$\Psi \in \C\B(\en)$
%bzgl.\ einer von Neumann-Algebra $\em$
%\glqq in der
``in the left variable".
%vorderen
%linken Variablen\grqq\ amplifizieren,
%indem wir alle obigen Schl\"usse -- {\textit{mutatis mutandis}} -- mit Tomiyamas rechter slice map
%anstelle der linken durchf\"uhren. Wir erhalten eine Amplifikation
%$\Chi_{\em, l}(\Psi) =:
This yields an amplification
$\id_{\em} \otp \Psi \in \C\B(\em \tt \en)$, which is completely determined by the equation:
\begin{eqnarray}
\langle \id_{\em} \otp \Psi, \rho \otp \tau \otp u \rangle = \langle \Psi, \tau \otp R_\rho(u) \rangle, \label{defr}
\end{eqnarray}
where $u \in \em \tt \en$, $\rho \in \em_*$ and $\tau \in \en_*$ are arbitrary.
Of course, $\id_{\em} \otp \Psi$ shares analogous properties to those of
$\Phi \otp \id_{\en}$.
\par
%%vgl mit klassischer ampl u vertauschung -- aber schon Tomiyamas art u witt's bzw mein
%%automat norm-resultat: vgl vertensorierten spnebel, eo ; u/o muss normal sein, wenn's kommutiert
%%intressant im Hinblick auf allgemeinen auto-normalit-satz!?, o nur gegenbsp: dh allg gehts nicht!: ??, eo
It is natural to ask for some relation between the operators in
$\C\B(\em \tt \en)$ which arise
%yielded by... !?, eo
by the two different kinds of amplification.
Let us briefly compare our situation with the classical setting of amplification of
operators on Hilbert spaces $\H$ and $\Ka$.
%an
%%operator
%s
%%on some Hilbert space $\H$
For operators $S \in \B(\H)$ and $T \in \B(\Ka)$, we consider the amplifications $S \otp \id_\Ka$ and
$\id_\H \otp T$ in $\B(\H) \tt \B(\Ka) = \B(\H \otp_2 \Ka)$. Then it is evident that we have
\begin{eqnarray}
(S \otp \id_\Ka) ~(\id_\H \otp T) = (\id_\H \otp T) ~(S \otp \id_\Ka), \label{commu}
\end{eqnarray}
and the tensor product of $S$ and $T$ is defined precisely to be this operator.
%concrete counterexample
%in forthcoming paper, hier wiho zitieren: wenn spiralnebel ex., immer, u
%ich zeige konkrete solche situationen!?, eo
%and this operator precisely enables one to define
%eind reinbringen!?, eo
\par
Back in our situation,
%if
for $\Phi \in \C\B(\em)$ and $\Psi \in \C\B(\en)$,
%are arbitrary,
the corresponding amplifications $\Phi \otp \id_\en$ and $\id_\em \otp \Psi$ will 
commute provided at least one of the involved mappings $\Phi$ or $\Psi$ is normal -- 
but not 
%commute 
in general. 
%We shall even prove 
In fact, we will even show that for most von Neumann algebras $\em$ and $\en$, 
$\Phi \in \C\B (\em)$ satisfying this innocent looking commutation relation for all $\Psi \in \C\B(\en)$ 
forces $\Phi$ to be normal! 
%%automatic normality!!: 
%%gibt auch Idee fuer Hof-witt--allggueltigkeit!?, eo 
%kommt charisierung von denen raus, wenn's klappt!?, eo 
This phenomenon, which may be surprising at first glance, 
is due to the fact that, in contrast to the multiplication \textit{in} a von Neumann algebra $\rr$ (in 
our case $\rr=\em \tt \en$),
the multiplication in $\C\B(\rr)$ is not $w^*$-continuous in both variables, but only in the left one. In fact,
the subset of $\C\B(\rr) \times \C\B(\rr)$ on which the product is $w^*$-continuous is
precisely $\C\B(\rr) \times \C\B^{\sigma}(\rr)$.
This is reflected in the following result which parallels relation (\ref{commu}) in our context.
\begin{satz} \label{commute}
Let ($\em$, $\en$) be an admissible pair.
If $\Psi \in \C\B^\sigma(\en)$ is a normal mapping, then we
%obtain
have for every $\Phi \in \C\B(\em)$:
$$(\Phi \otp \id_\en) ~(\id_\em \otp \Psi) = (\id_\em \otp \Psi) ~(\Phi \otp \id_\en).$$
%was bei normalit des andern?!: dasselbe!?, vgle ueberall!?, eo
\end{satz}
%
%
%kommen im klassischen fall due to the w*-cont in each var!?;
%kann man aus einseitiger stetkeit was rausholen!?, eo
%vgl top Z_t
\begin{proof}
Fix $u \in \em \tt \en$, $\rho \in \em_*$ and $\tau \in \en_*$.
We first
%note
remark that
\begin{eqnarray}
\langle \id_{\em} \otp \Psi, \rho \otp \tau \otp u \rangle &\stackrel{(\ref{defr})}{=}& \langle \Psi, \tau \otp R_\rho(u) \rangle
\nonumber\\
&=& \langle \Psi(R_\rho(u)), \tau \rangle \nonumber\\
&=& \langle R_\rho(u), \Psi_*(\tau) \rangle \nonumber\\
&=& \langle u, \rho \otp \Psi_*(\tau) \rangle, \label{urops}
\end{eqnarray}
where $\Psi_*$ denotes the pre-adjoint mapping of $\Psi$.
\par
To establish the claim, it suffices to prove that:
$$\langle ( \Phi \otp \id_\en ) ~( \id_\em \otp \Psi ), \rho \otp \tau \otp u \rangle =
\langle ( \id_\em \otp \Psi ) ~( \Phi \otp \id_\en ), \rho \otp \tau \otp u \rangle.$$
The left side takes on the following form:
\begin{eqnarray*}
\langle ( \Phi \otp \id_\en ) ~( \id_\em \otp \Psi ), \rho \otp \tau \otp u \rangle &=&
\langle \Chi_\en(\Phi), \rho \otp \tau \otp ( \id_\em \otp \Psi )(u) \rangle \\
&=& \langle \Phi, \rho \otp L_\tau [( \id_\em \otp \Psi )(u)] \rangle,
\end{eqnarray*}
whereas on the right side, we obtain:
\begin{eqnarray*}
\langle ( \id_\em \otp \Psi ) ~( \Phi \otp \id_\en ), \rho \otp \tau \otp u \rangle &=&
\langle ( \id_\em \otp \Psi ) ~( \Chi_\en(\Phi) (u) ), \rho \otp \tau \rangle \\
&\stackrel{(\ref{urops})}{=}& \langle \Chi_\en(\Phi) (u), \rho \otp \Psi_*(\tau) \rangle \\
&=& \langle \Chi_\en(\Phi), \rho \otp \Psi_*(\tau) \otp u \rangle \\
&=& \langle \Phi, \rho \otp L_{\Psi_*(\tau)} (u) \rangle.
\end{eqnarray*}
Hence, it remains to be shown that
$$L_\tau [( \id_\em \otp \Psi )(u)] = L_{\Psi_*(\tau)} (u).$$
But for arbitrary $\rho' \in \em_*$, we get:
\begin{eqnarray*}
\langle L_\tau [( \id_\em \otp \Psi )(u)], \rho' \rangle &=& \langle ( \id_\em \otp \Psi )(u), \rho' \otp \tau \rangle \\
&\stackrel{(\ref{urops})}{=}& \langle u, \rho' \otp \Psi_*(\tau) \rangle \\
&=& \langle L_{\Psi_*(\tau)} (u), \rho' \rangle,
\end{eqnarray*}
which yields the claim.
\end{proof}
We now come to the main result of this section, which establishes a converse of the above Theorem 
for a very wide class of von Neumann algebras. It may be interpreted as a result on ``automatic normality'', i.e., 
some simple, purely algebraic condition -- namely commuting of the associated amplifications -- automatically implies the 
strong topological property of normality. Besides one 
%extremely weak 
very mild technical condition, the first von Neumann algebra, 
$\em$, can be completely arbitrary, the second one, $\en$, is only 
%supposed 
assumed to be properly infinite; we will show by an 
example that 
%in general, 
without the latter condition, the conclusion does not hold in general. 
%As for the 
The weak technical assumption mentioned says, roughly speaking, that the von Neumann algebras involved should not 
be acting on Hilbert spaces of pathologically large dimension. We remark that for separably acting 
von Neumann algebras $\em$ and $\en$, or, more generally, in case $\em \tt \en$ is countably decomposable, 
our assumption is trivially satisfied. 
%To be more precise, we will exclude 
%
In order to formulate 
%the theorem, we 
%introduce 
%our result 
the condition precisely, 
we use the following natural terminology which has been introduced in \cite{ich1}. 
\begin{defi} \label{ichhu} 
Let $\rr \subseteq \B(\H)$ be a von Neumann algebra. Then we define the $\mathrm{de\-com\-po\-sa\-bi\-lity}$ 
$\mathrm{number}$ of $\rr$, 
denoted by $\dec(\rr)$, to be the 
%largest 
smallest cardinal number $\kappa$ such that 
%there exists 
every family of non-zero pairwise 
orthogonal projections in $\rr$ 
%whose 
%with 
has at most cardinality 
%is 
$\kappa$. 
\end{defi} 
\begin{bem} 
Of course, a von Neumann algebra $\rr$ is countably decomposable if and only if $\dec(\rr) \leq \aleph_0$. 
\end{bem} 
%\par 
We are now ready to state the main theorem of this section. 
\begin{satz} \label{umk} 
Let $\em$ and $\en$ be von Neumann algebras such that $\dec (\em \tt \en)$ is a non-measurable cardinal, 
and suppose $\en$ is properly infinite. If $\Phi \in \C\B(\em)$ is such that 
$$(\Phi \otp \id_\en) ~(\id_\em \otp \Psi) = (\id_\em \otp \Psi) ~(\Phi \otp \id_\en)$$
for all $\Psi \in \C\B(\en)$, then $\Phi$ must be normal. 
\end{satz} 
\begin{bem} 
The assumption that the decomposability number of $\em \tt \en$ be non-measurable, is very weak and in fact just excludes 
a set-theoretic pathology. We briefly recall that a cardinal $\kappa$ is said to be (real-valued) measurable if for every 
set $\Gamma$ of cardinality $\kappa$, there exists a diffused probability measure on the power set 
$\mathfrak{P}(\Gamma)$. -- Measurability is a property of ``large'' cardinals ($\aleph_0$ is of course not measurable). 
%Of course, $\aleph_0$ is not measurable; Measurable cardinals are 
%At this point, we content ourselves to note 
In order to demonstrate how weak the restriction to non-measurable cardinals is, let us just mention the fact (cf.\ \cite{kama}, 
\S 1, p.\ 106, 108) that 
the existence of measurable cardinals cannot be proved in ZFC (= the axioms of Zermelo--Fraenkel + the axiom of choice), 
and that it is consistent with ZFC to assume the non-existence of measurable cardinals (\cite{hand}, \S 4, Thm.\ 4.14, p.\ 972). 
For a further discussion, we refer to \cite{ich1}, and the references therein. 
\end{bem} 
We begin now with the preparations needed for the proof of Theorem \ref{umk}. 
We first note that, even though 
%having 
requiring a non-measurable decomposability number means only excluding exotic von Neumann algebras, it is already enough 
to ensure a very pleasant property of the predual. The latter 
%which 
%it 
is actually of 
%fundamental 
great (technical) importance in the proof of Theorem \ref{umk} and is therefore stated explicitly in the following. 
\begin{prop} \label{mazur} 
Let $\rr$ be a von Neumann algebra. 
%with 
Then $\dec(\rr)$ is non-measurable if and only if the predual $\rr_*$ has Mazur's property (i.e., 
$w^*$-sequentially continuous functionals on $\rr$ are $w^*$-continuous -- and hence belong to $\rr_*$). 
%decomposability number
\end{prop}
\begin{proof} 
This is 
%part of 
Thm.\ 3.11 in \cite{ich1}. 
\end{proof}
The crucial idea in the proof of Theorem \ref{umk} is to use a variant of 
the concept of a \textit{countable spiral nebula}, which has been introduced in \cite{wiho}, \S 1.1 (cf.\ ibid., Lemma 1.5). 
\begin{defi} \label{nebel} 
Let $\rr$ be a von Neumann algebra. A sequence $(\al_n , e_n)_{n \in \N}$ of ${}^*$-automorphisms $\al_n$ on $\rr$ and 
projections $e_n$ in $\rr$ will be called a $\mathrm{countable ~reduced ~spiral ~nebula}$ on $\rr$ if the following holds true: 
$$e_n \leq e_{n+1} , ~{\mathrm{WOT}}-\lim_n e_n = 1 , ~ \al_m (e_m) \perp \al_n (e_n) ~\mathrm{for ~all}~ m,n \in \N_0 , 
m \not= n .$$ 
\end{defi} 
The following Proposition is the crucial step in order to establish Theorem \ref{umk}. It shows 
that a countable reduced spiral nebula may be used to derive a result on automatic normality -- and not 
only on automatic boundedness, as is done in \cite{wiho}, Lemma 1.5. 
%%Thm 3.11!! 
\begin{prop} \label{hauprop} 
Let $\rr$ be a von Neumann algebra such that $\dec(\rr)$ is non-measurable, and suppose that there is 
a countable reduced spiral nebula $(\al_n , e_n)$ on $\rr$. 
\begin{itemize} 
\item[(i)] Put $l_k \dogl \frac{k (k+1)}{2}$, and let $\mathfrak{F}$ be a free ultrafilter on $\N$. Then the 
operators 
$$\Psi_n \dogl w^*-\lim_{k \rightarrow \mathfrak{F}} \al^{-1}_{l_k + n}$$
are completely positive and unital. 
\item[(ii)] If $\Phi \in \B(\rr)$ commutes with all elements of the set 
$S \dogl \overline{\{ \al_n^{-1} \mid n \in \N_0 \}}^{w^*} \subseteq \C\P(\rr)$, 
then $\Phi$ is normal. 
\end{itemize}
\end{prop} 
\begin{proof} 
The statement (i) is clear. In order to prove (ii), 
%since $\dec(\rr)$ is non-measurable, 
in view of Proposition \ref{mazur}, we only have to show that 
if $(x_n)_{n \in \N_0} \subseteq \Ball (\rr)$ converges $w^*$ to $0$, then the same is true for the sequence $(\Phi (x_n))_n$. 
It suffices to prove that if $(\Phi (x_{n_i}))_i$ is a $w^*$-convergent subnet, then its limit must be $0$. 
\par 
The projections $\al_n (e_n)$ being pairwise orthogonal, noting that $l_{\kappa +1} = l_\kappa + \kappa +1$, we see that 
$$\widetilde{x} \dogl \sum_{\kappa=0}^{\infty} \sum_{\nu = 0}^{\kappa} 
\al_{l_\kappa + \nu} (e_{l_\kappa + \nu} x_\nu e_{l_\kappa + \nu})  \in \rr .$$ 
%As in 
Now we deduce (cf.\ the proof of \cite{wiho}, Lemma 1.5) 
%it now follows 
%we deduce 
that 
\begin{equation} 
\Psi_n (\widetilde{x}) = x_n ~\mathrm{for ~all}~ n \in \N_0 . \label{faktori} 
\end{equation} 
%In order to keep our exposition self-contained, we reproduce the short argument here. -- For $$
Let $\Psi \dogl w^*-\lim_{j} \Psi_{n_{i_{j}}} \in \Ball(\C\B(\rr))$ be a $w^*$-cluster point of the net $(\Psi_{n_{i}})_i$. 
Then $\Psi \in S$, and we obtain, using (\ref{faktori}): 
\begin{eqnarray*} 
\lim_i \Phi (x_{n_i}) &=& \lim_i \Phi (\Psi_{n_i} (\widetilde{x})) 
= \lim_i \Psi_{n_i} (\Phi (\widetilde{x})) \\ 
&=& \lim_j \Psi_{n_{i_{j}}} (\Phi (\widetilde{x})) 
= \Psi (\Phi (\widetilde{x})) \\ 
&=& \Phi (\Psi (\widetilde{x})) 
= \Phi (\lim_j \Psi_{n_{i_{j}}} (\widetilde{x})) \\ 
&=& \Phi (\lim_j x_{n_{i_{j}}}) 
= \Phi (0) = 0 , 
\end{eqnarray*} 
which finishes the proof. 
\end{proof} 
We are now sufficiently prepared for Theorem \ref{umk}. 
\begin{proof}
Since $\en$ is properly infinite, it is 
%spatially 
isomorphic to $\en \tt \B(\ell_2(\zett))$; cf., e.g., Appendix C, ``Theorem'', in \cite{vand}. By applying the argument in the first 
part of the proof of Prop.\ 2.2 in \cite{wiho}, we 
%see that 
%there exists 
can construct a countable reduced spiral nebula $(\al_n , e_n)$ on $\B(\ell_2(\zett))$. Hence, 
$(\widetilde{\al}_n , \widetilde{e}_n) \dogl (\id_\em \otimes \id_\en \otimes \al_n , 1_\em \otimes 1_\en \otimes e_n)$ 
is a countable reduced spiral nebula on $\em \tt \en \tt \B(\ell_2(\zett)) = \em \tt \en$. 
We will now apply Proposition \ref{hauprop} to the von Neumann algebra $\em \tt \en$ and the mapping 
$\Phi \otimes \id_\en \in \C\B(\em \tt \en)$ to show that the latter, and hence $\Phi$ itself, is normal. -- To this end, 
we only have to prove that $\Phi \otimes \id_\en$ commutes with all elements of the set 
$S \dogl \overline{\{ \widetilde{\al}_n^{-1} \mid n \in \N_0 \}}^{w^*}$. But the amplification in the first variable, $\Chi_\em$, 
is $w^*$-$w^*$-continuous (this is established in the same fashion as we did for $\Chi_\em$ in Theorem \ref{amsatz}), 
and this entails that every element of $S$ is of the form $\id_\em \otimes \Psi$ for some $\Psi \in \C\B(\en)$. 
%using property (iii) (in Definition \ref{amdef}) 
%established for our amplification 
\end{proof} 
\begin{bem} 
The assumption of $\en$ being properly infinite 
%ensures a certain richness 
may be interpreted as a certain richness condition which is necessary to be imposed on the second von Neumann algebra. 
Indeed, if in contrast $\en$ is, e.g., a finite type I factor, the statement is wrong in general. This is easily seen as follows: 
In this case, $\en = \B(\ell_2^n) = M_n$ for some $n \in \N$, and hence $\C\B(\en) = (\Te (\ell_2^n) \tp \B(\ell_2^n))^* 
= (\Te (\ell_2^n) \tp \Ka(\ell_2^n))^* = \C\B^\sigma (\en)$. 
Now let $\em$ be an arbitrary von Neumann algebra such that the singular part of the Tomiyama--Takesaki decomposition of $\C\B(\em)$ is 
non-trivial, i.e., $\C\B^{s}(\em) \not= (0)$. Then by Theorem \ref{commute}, every $\Phi \in \C\B^{s}(\em)$ will satisfy 
the commutation relation in Theorem \ref{umk} for all $\Psi \in \C\B(\en)=\C\B^\sigma (\en)$, even though it is singular. 
\end{bem}
%\par 
We finish our discussion with 
%let now be 
considering again an arbitrary admissible pair ($\em$, $\en$). 
In view of Theorem \ref{commute}, it would be natural, following the model of the tensor product of bounded operators on
Hilbert spaces, to define a tensor product of
%of completely bounded mappings
$\Phi \in \C\B(\em)$ and
$\Psi \in \C\B^{\sigma}(\en)$ by setting:
\begin{eqnarray*}
\Phi \overline{\otimes} \Psi \dogl (\Phi \otp \id_{\en}) ~(\id_{\em} \otp \Psi) = (\id_{\em} \otp \Psi) ~(\Phi \otp \id_{\en}),
\end{eqnarray*}
which defines an operator in $\C\B(\em \tt \en)$. It would be interesting to go even further and consider, for arbitrary
$\Phi \in \C\B(\em)$ and $\Psi \in \C\B(\en)$, the
two ``tensor products'' $\Phi \widetilde{\otimes}_1 \Psi$ and $\Phi \widetilde{\otimes}_2 \Psi$, defined respectively
through
$$\Phi \widetilde{\otimes}_1 \Psi \dogl (\Phi \otp \id_{\en}) ~(\id_{\em} \otp \Psi) \in \C\B(\em \tt \en)$$
and
$$\Phi \widetilde{\otimes}_2 \Psi \dogl (\id_{\em} \otp \Psi) ~(\Phi \otp \id_{\en}) \in \C\B(\em \tt \en).$$
By Theorem \ref{umk}, we know that these are different in general, even in the case of von Neumann algebras $\em$ and $\en$. 
%At this point, we would only like to 
We wish to close this section by stressing the resemblance of
%this construction
these considerations to the well-known construction of the two Arens products on the bidual of a Banach algebra, which in general
are different. From this point of view, the subalgebra $\C\B^{\sigma}(\en)$ could play the role of what in the context
of Banach algebras is known as the topological centre (for the latter term, see, e.g., \cite{dales}, Def.\ 2.6.19). 
%%%-- and Theorem says that it is precisely CB^sigma!?, eo !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 
%
%entsprechende Eigenschaften besitzt.
%wie eigen genau!?
%wie mit eind., oder egal!?
%
%
%\par
%Diese Untersuchung legt es nahe, nach dem Vorbild des Tensorproduktes von beschr\"ankten
%Operatoren auf Hilbertr\"aumen nun ein Tensorprodukt von vollst\"andig beschr\"ankten
%Abbildungen auf von Neumann-Algebren zu erkl\"aren. -- Sind $\Phi \in \C\B(\em)$, $\Psi \in \C\B(\en)$, so m\"ochten wir
%in Analogie zum von Neumann-Tensorprodukt nun
%das Tensorprodukt\footnote{(von Neufang)} $\Phi \overline{\otimes} \Psi \in \C\B(\em \tt \en)$ definieren verm\"oge:
%$$\Phi \overline{\otimes} \Psi \dogl
%\Chi_{\en, r}(\Phi) ~\Chi_{\em, l}(\Psi) =
%(\Phi \otp \id_{\en}) ~(\id_{\em} \otp \Psi).$$
%wobei wir beachten, da"s der rechtsstehende Ausdruck ein Element in $\C\B(\em \overline{\otimes} \en)$ definiert.
%\par
%Aufs Neue best\"atigte dies unsere These, im Rahmen der Theorie
%{\textit{Operatorr\"aume -- vollst\"andig beschr\"ankte Abbildungen}}
%-Theorie
%nehme
%spiele die Algebra $\C\B(\B(\H))$
%den Platz ... ein.
%die Rolle, welche $\B(\H)$ in der Theorie
%{\textit{von Neumann-Algebren -- beschr\"ankte Operatoren}} \"ubernimmt.
%Anstelle des von Neumann-Tensorproduktes $\B(\H) \tt \B(\Ka)$
%daher sollte es schwacher abschluss sein!, vgl auch kraus' sigma-w-closed-Sachen! eeeeeeooooooo  - was meinte witt!?, eeoo
%tr\"ate nun ein entsprechendes schwaches Tensorprodukt von $\C\B(\B(\H))$ und $\C\B(\B(\K))$ -- oder allgemeiner
%von $\C\B(\em)$ und $\C\B(\en)$, wobei $\em$ und $\en$ von Neumann-Algebren sind.
%\par
%Dies mag den Beginn einer Tensorprodukt-Theorie auf dem \glqq nichtkommutativen\grqq\ Niveau der isometrisch in $\C\B(\B(\H))$
%anstatt (wie in der klassischen Theorie der Operatoralgebren) isometrisch in $\B(\H)$ dargestellten Algebren einl\"auten.
%Unter diesen finden sich ja, wie dem Leser nunmehr sehr wohl bewu"st, Algebren, die ebenso interessant wie
%im Rahmen der klassischen Theorie schwer zug\"anglich sind -- Duale von $C^*$-Algebren wie $\M(\G)$ oder $\lu$.
%\par
%Algebren,
%die solch eine Darstellung zulassen, m\"ochten wir vision\"ar als {\textit{$\C\B$-Algebren}} bezeichnen. Sie harren freilich
%noch einer axiomatischen Beschreibung, wie sie bei $C^*$-Algebren oder Operatorr\"aumen\footnote{im letzteren Falle
%\"uber den mittlerweile klassischen Satz von Ruan} wohlbekannt ist. Doch ist unbestritten die
%Theorie der Tensorprodukte im Studium klassischer Operatoralgebren wie auch von Operatorr\"aumen eines, wenn nicht
%{\textit{das}} zentrale Hilfsmittel, die Struktur der Objekte zu fassen; dies sollte im Falle der $\C\B$-Algebren
%freilich
%nicht anders sein.
% m\"ochten
%Eine Theorie der Tensorprodukte von $\C\B$-Algebren wiederum sollte aber nat\"urlich auf einem Tensorprodukt der Algebren
%$\C\B(\em)$ und $\C\B(\en)$, wobei $\em$ und $\en$ von Neumann-Algebren sind, fu"sen; daher ist eventuell mit den
%obigen Ergebnissen ein erster, kleiner Schritt getan. Wir betonen abschlie"send, da"s auf dem Hintergrund
%dieses Vorschlages einer Tensorprodukt-Theorie die {\textit{vollst\"andige}} Beschr\"anktheit der betreffenden Abbildungen
%freilich
%essentiell ist, wie ja schon f\"ur die Bildung unserer Amplifikation. Hieraus erhellt, um ein
%letztes Mal auf unsere Darstellungen zu sprechen zu kommen, die besondere Bedeutung der Algebren $\gi(\M(\G))$ und
%$\gu(\lu)$ als Unteralgebren von $\C\B^\sigma(\B(\H))$ bzw.\ $\C\B(\B(\H))$, $\H=L_2(\G)$, im
%Gegensatz
%Vergleich zu den
%$p$-Varianten $\gip(\M(\G))$ und $\gup(\lu)$.
%\par
%Im Rahmen einer Theorie der $\C\B$-Algebren sind also $\gi(\M(\G))$ und
%$\gu(\lu)$ \glqq lediglich\grqq\ Beispiele,
%wenn auch
%doch dank ihrer banachalgebraischen Komplexit\"at \"au"serst interessante. Sie zeigen,
%da"s die zentralen Fragestellungen aus der klassischen Theorie der Operatoralgebren -- wie Bikommutantenprobleme (!) --
%auch im Rahmen der $\C\B$-Algebren in hochgradig nichttrivialen F\"allen und zudem in h\"ochstm\"oglicher Allgemeinheit
%positive Antworten zulassen. Dies zu demonstrieren, war der vorliegenden Arbeit ein Hauptanliegen.
%%zugleich satruktur-bezug wie immer bei darst-theorie!!?, eeoo; bezug zu vorher, nachher, vorwort, eeeooo !!!??
%--
%-- Doch
%wollen wir
%sollte der
%dieser Beginn
%den
%Beginn einer solchen {\textit{allgemeinen}} Darstellungstheorie, insbesondere der zugeh\"origen Theorie der Tensorprodukte,
%mit der Vorstellung einer abstrakten Theorie der $\C\B$-Algebren
%im Rahmen der vorliegenden Arbeit lediglich in einem Ausblick leicht an- und nun ausklingen.
%
%
%\section{An approach to a theorem of May-Neuhardt-Wittstock via slice maps}
%\section{A slice map approach to a theorem of May, Neuhardt and Wittstock}

\section{Uniquenes of the amplification and a generalization 
of the Ge--Kadison Lemma} \label{ge} 

%\par 
In the following, we 
%shall be interested in giving a positive answer to 
will show that under different natural conditions, 
%the uniqueness problem has a positive answer 
an algebraic amplification is uniquely determined. 
%in the sense of 
%. We 
%present 
%show that under various different 
%will see that 
%under different natural conditions. 
%, 
%which ensure uniqueness of 
%the amplification indeed is unique. 
This 
leads in particular to a remarkable generalization of the so-called Ge--Kadison Lemma 
(see Proposition \ref{ichgekad}), which is obtained as an application of our Theorem \ref{commute}. 
\par 
But before establishing positive results, we briefly point out why 
%obviously 
in general, an algebraic amplification 
is highly non-unique. In \cite{tom}, p.\ 28, Tomiyama states, in the context of von Neumann 
algebras, that the ``product projection'' of two 
projections of norm one ``might not be unique'' (of course, in our case, the second projection of norm one 
would be the identity mapping). The following simple argument shows that whenever $\em$ and $\en$ 
are infinite-dimensional dual operator spaces, there are uncountably many different algebraic amplifications 
of a map $\Phi \in \C\B(\em)$ to a completely bounded map on $\em \tt \en$, regardless of $\Phi$ being normal or not. 
Namely, for any non-zero functional 
$\varphi \in ( \em \tt \en )^*$ which vanishes on $\em \ti \en$, and any non-zero vector $v \in \em \tt \en$, 
$$\Chi_{\en}^{\varphi, v} (\Phi) \dogl \Chi_{\en} (\Phi) - \langle \varphi , \Chi_{\en} (\Phi) ( \cdot ) \rangle ~v$$ 
defines such an algebraic amplification. 
Furthermore, the amplification $\Chi_{\en}^{\varphi, v}$ can of course be taken arbitrarily close to an isometry: 
for every $\varepsilon > 0$, an obvious choice of $\varphi$ and $v$ yields an algebraic amplification 
$\Chi_{\en}^{\varphi, v} : \C\B (\em) \longrightarrow \C\B ( \em \tt \en )$ such that 
$$(1-\varepsilon ) ~\| \Phi \|_{\mathrm{cb}} \leq \| \Chi_{\en}^{\varphi, v} (\Phi) \|_{\mathrm{cb}} 
\leq (1+\varepsilon ) ~\| \Phi \|_{\mathrm{cb}}$$ 
for all $\Phi \in \C\B(\em)$. 
\par 
We 
%first 
begin our investigation by noting 
%the following 
%easy 
a criterion which establishes the uniqueness of amplification 
%of the latter 
for a large class of
von Neumann algebras, 
%under .
though only assuming part of the requirements listed in Definition \ref{amdef}. 
%sagen: kann man nicht immer erwarten!!
\begin{prop} \label{eindam}
Let $\em$ and $\en$ be von Neumann algebras, where $\em$ is an injective factor.
%Dann ist
If $\Chi': \C\B(\em) \longrightarrow \C\B(\em \tt \en)$ is a bounded linear mapping 
%it is uniquely determined by
which satisfies the algebraic amplification condition (AAC) and properties
(iii) and (iv) in Definition \ref{amdef}, then $\Chi' = \Chi_\en$.
\end{prop}
\begin{proof}
%Let
%nun
%$\Chi'$ be another amplification.
Owing to \cite{chsm}, Thm.\ 4.2, we have the density relation:
$${\overline{\C\B^{\sigma}(\em)}}^{w^*} = \C\B(\em).$$
%wann (genau) cb = cb^sigma?, wie mit Invarianz von
%Sei nun $\Chi'$ eine weitere Amplifikation.
%Dann
%Demnach
Hence, due to 
%Thanks to the
their $w^*$-$w^*$-continuity (condition (iii)), $\Chi_\en$ and $\Chi'$ 
%, both mappings 
will be identical if they only coincide
on $\C\B^{\sigma}(\em)$. But if $\Phi \in \C\B^{\sigma}(\em)$, by condition (iv), the mappings
$\Chi_\en (\Phi)$ and $\Chi'(\Phi)$
are normal, so that
%they are equal
to establish their equality, it suffices to show that they coincide on all elementary tensors $S \otp T \in \em \tt \en$.
But this is true since they both fulfill condition 
%by
%nach
(AAC). 
\end{proof} 
%spaeter sagen: analog dazu geschlossen!
%Bislang haben wir uns wohlweislich \"uber die Frage ausgeschwiegen, ob
%es in der allgemeinen Situation von Definition
%\label{}
%denn \"uberhaupt eine Amplifikation im Sinne von Definition \ref{amdef} in der allgemeinen Situation beliebiger von
%Neumann-Algebren $\em$ und $\en$ und -- nicht notwendig normaler (!) -- vollst\"andig beschr\"ankter Abbildungen $\Phi$
%existiert. Dieses Resultat ist bislang unbekannt, vielmehr aber unser Haupt-
%Schr\"anken wir
%\par
%Etwas genauer gesprochen gilt nun:
%We now come to our main result.
%
We will now present our generalization of the Ge--Kadison Lemma which 
%in its original form first appeared in 
was first obtained 
%in its original form 
in \cite{gekad} 
(ibid., Lemma F) and used in the 
%framework of the 
solution of the splitting problem for tensor products of von Neumann algebras (in the factor case). 
%Lemma F in Ge-Kad 
Furtheron, a version of the Lemma was again used by Str\u{a}til\u{a} and Zsid\'{o} in \cite{stzs} in order to 
derive an extremely general commutation theorem which extends both Tomita's classical 
commutant theorem and the above mentioned splitting result (\cite{stzs}, Thm.\ 4.7). 
The result we give below generalizes the Lemma in the version as stated (and proved) in \cite{stzs}, \S 3.4, which 
the authors refer to as a ``smart technical device''. In \cite{stzs}, it is applied to \textit{normal} 
conditional expectations of von Neumann algebras (ibid., \S 3.5); of course, these are in particular 
\textit{completely positive}. Our generalization yields an analogous result 
which in turn can be applied 
%, in particular, 
to \textit{arbitrary} completely bounded mappings. This fact may be of interest in the further development 
of the subject treated in \cite{gekad} and \cite{stzs}. Furthermore, 
we prove that the Lemma not only holds for 
%extends the framework of 
von Neumann algebras 
%to 
but for arbitrary admissible pairs, a situation which is not considered in 
the latter articles. Hence, our version is likely to prove useful in the interesting problem 
%It is of course interesting to ask for 
of obtaining a splitting theorem 
in the more general context of $w^*$-spatial tensor products of 
ultraweakly closed subspaces -- a question which is actually hinted at in 
%the 
%introduction of 
\cite{gekad} (p.\ 457). 
% mentioned. 
%
%conditional expectations
%%It seems interesting 
%In the following, we shall refer to the version of the Lemma stated (and proved) in \cite{stzs}, \S 3.4, which 
%the authors 
%of \cite{stzs} 
%who 
%refer to the Lemma as 
%call a ``smart technical device''
%We shall refer below to the version of the Lemma 
\begin{prop} \label{ichgekad} 
Let $(\em , \en)$ be an admissible pair, and $\Phi \in \C\B(\em)$ an arbitrary completely bounded operator. 
Suppose $\Theta : \em \tt \en \longrightarrow \em \tt \en$ is $\mathrm{any}$ map 
%satisfying 
which satisfies, for some non-zero element $n \in \en$: 
\begin{itemize} 
\item[(i)] $\Theta$ commutes with the 
slice maps $\id_\em \otimes \tau n$ ($\tau \in \en_*$) 
\item[(ii)] $\Theta$ coincides with $\Phi \otimes \id_\en$ on $\em \otimes n$. 
\end{itemize} 
Then we must have $\Theta = \Phi \otimes \id_\en$. 
\par 
In particular, if $\Phi \in \C\B(\em)$ and $(\Phi_\al)_\al \subseteq \C\B(\em)$ is a bounded net converging $w^*$ to $\Phi$, 
%\stackrel{w^*}{\longrightarrow} \Phi$, where $\Phi_\al$, $\Phi \in \C\B(\em)$, 
then 
$\Phi_\al \otimes \id_\en \stackrel{w^*}{\longrightarrow} \Phi \otimes \id_\en$. 
\end{prop}
\begin{bem} 
%\newline 
%\begin{itemize} 
%\item[(i)] 
(i) The known version of the Lemma supposes that $\em$ and $\en$ are von Neumann algebras, and that 
the mapping $\Phi$ is normal and completely positive. As shown by the above, all these 
assumptions can be either weakened or even just dropped. 
%\item[(ii)] 
\par 
(ii) The last statement in the Proposition is even true $\mathrm{without}$ assuming the net $(\Phi_\al)_\al$ 
to be bounded, and in this generality follows directly from 
%the above 
our Theorem \ref{amsatz} (property (iii) in Definition \ref{amdef}). Nevertheless, this ``unbounded'' version 
%is 
does not appear in the literature. (We remark that the assumption of boundedness is missing in the statement of Lemma 3.4 in 
\cite{stzs}, though needed in the proof presented there.) 
%Nevertheless, w
We shall 
%briefly 
point out below 
how the ``bounded'' version can be easily obtained by using the first part of the Proposition (cf.\ \cite{stzs}, \S 3.4). 
%\end{itemize} 
\end{bem} 
\begin{proof} 
Using the first result of the last section, our argument follows the same lines as the one given in \cite{stzs}; we 
present the proof because of its brevity. -- Fix a non-zero element $m \in \em$. We obtain for every 
$u \in \em \tt \en$, $\rho \in \em_*$ and $\tau \in \en_*$: 
\begin{eqnarray*} 
\langle \Theta (u), \rho \otimes \tau \rangle ~m \otimes n &=& 
\left[ (\rho m \otimes \id_\en ) ~( \id_\em \otimes \tau n ) ~\Theta  \right] (u) \\ 
&=& \left[ (\rho m \otimes \id_\en ) ~\Theta ~( \id_\em \otimes \tau n )  \right] (u) \quad \mathrm{by ~(i)} \\ 
&=& \left[ (\rho m \otimes \id_\en ) ~(\Phi \otimes \id_\en ) ~( \id_\em \otimes \tau n )  \right] (u) \quad \mathrm{by ~(ii)} \\ 
&=& \left[ (\rho m \otimes \id_\en ) ~( \id_\em \otimes \tau n ) ~(\Phi \otimes \id_\en ) \right] (u) 
\quad \mathrm{by ~Theorem ~\ref{commute}} \\ 
&=& \langle (\Phi \otimes \id_\en) (u), \rho \otimes \tau \rangle ~m \otimes n , 
\end{eqnarray*} 
which proves the first statement in the Proposition. 
%Since closed balls in $\C$
In order to prove the second assertion, we only have to show that if $\Theta$ is any $w^*$-cluster point of the net 
$(\Phi_\al \otimes \id_\en)_\al$, then $\Theta = \Phi \otimes \id_\en$. But $\Theta$ trivially satisfies condition (ii) 
in our Proposition, and using the normality of the mappings $\id_\em \otimes \tau n$, 
another application of Theorem \ref{commute} shows that $\Theta$ also meets condition (i), whence the first part of the 
Proposition finishes the proof. 
\end{proof} 



\section{Completely bounded module homomorphisms and Tomiyama's slice maps} \label{sechs}
%Approach to a thm of MNW
%In this section, $\em$ and $\en$ will always denote arbitrary von Neumann algebras.
The main objective of \cite{mnw} was to establish the following remarkable property of the amplification
mapping
$$\C\B(\B(\H)) \ni \Phi \mapsto \Phi^{(\infty)} \in \C\B(\B(\H) \tt \B(\K)).$$
If $\rr \subseteq \B(\H)$ is a von Neumann subalgebra, and if $\Phi \in \C\B_{\rr}(\B(\H))$ is an
$\rr$-bimodule homomorphism, then $\Phi^{(\infty)}$ is an $\rr \tt \B(\K)$-bimodule homomorphism, i.e.,
$\Phi^{(\infty)} \in \C\B_{\rr \tt \B(\K)}(\B(\H) \tt \B(\K))$; see \cite{mnw}, Prop.\ 2.2 (and
\cite{bim}, Thm.\ 4.2, for a
%more
generalization).
% situation.
\par
An important special
%case
situation, which nevertheless shows the 
%full 
strength and encodes the essential statement of the result, is the case where 
$\rr = {\mathbb{C}} ~\1_{\B(\H)}$. Then the above theorem reads as follows:
\par
$(**) ~{}$ Every $\Phi \in \C\B(\B(\H))$
%admits
%an amplification
can be amplified to an
%completely bounded
$\1_{\B(\H)} \otimes \B(\K)$-bimodule homomorphism $\Phi^{(\infty)} \in \C\B(\B(\H) \tt \B(\K))$.
\par
The aim of this section is to prove
an analogous
%the corresponding
result where now $\B(\H)$ and $\B(\K)$ are replaced by arbitrary
von Neumann algebras $\em$ and $\en$, using our amplification $\Chi_\en$. In this section, $\em$ and $\en$ will
always denote von Neumann algebras.
\par
The proofs of \cite{mnw}, Prop.\ 2.2 and \cite{bim}, Thm.\ 4.2
%both
use either the Wittstock decomposition theorem or
%the
a corresponding result by Paulsen (\cite{paul}, Lemma 7.1) to reduce to the case of a completely positive mapping,
where the assertion is then obtained by a fairly subtle analysis involving the Schwarz inequality.
In contrast to this,
%our
the explicit form of $\Chi_\en$ provides us with a
%straightforward approach
%easy and elegant!?, eo
direct route to
%our
the above announced goal; indeed, as we shall see,
%we can
the
%assertion
statement asserting the module homomorphism property of the amplified mapping
reduces to a corresponding statement about Tomiyama's slice maps which in turn can be verified in an elementary fashion (see
Lemma \ref{easy} below).
\par
In order to
%formulate
prove our result, a little preparation is needed.
In the
%following
sequel, we restrict ourselves to left slice maps; analogous statements hold of course in the case of
right slice maps. -- As is well-known (and easy to see), Tomiyama's left slice maps satisfy the following module
homomorphism property:
$$L_\tau ( (a \otimes \1_\en) u (b \otimes \1_\en) ) = a L_\tau (u) b,$$
where $u \in \em \tt \en$, $a, b \in \em$, $\tau \in \en_*$.
In the
%The
following (elementary) lemma, we shall
establish an analogous property involving elementary tensors of the form $\1_\em \otimes a$ and $\1_\em \otimes b$ for
$a, b \in \en$. To this end, we briefly recall that for every von Neumann algebra $\en$, the predual $\en_*$
is an $\en$-bimodule in a very natural manner, the actions being given by
$$\langle \tau \cdot a, b \rangle = \langle \tau, ab \rangle \quad \mathrm{and} \quad
\langle a \cdot \tau , b \rangle = \langle \tau, ba \rangle,$$
%where
for $a, b \in \en$, $\tau \in \en_*$.
%Using this notation, we have
For later purposes, we note the (easily verified) equation:
\begin{eqnarray}
(\1_\em \otimes a) \cdot (\rho \otimes \tau) \cdot (\1_\em \otimes b) = \rho \otimes (a \cdot \tau \cdot b), \label{gleichun}
\end{eqnarray}
where $a, b \in \en$, $\rho \in \em_*$, $\tau \in \en_*$.
We now come to
\begin{lemma} \label{easy}
Let $\em$ and $\en$ be von Neumann algebras. Then Tomiyama's left slice map satisfies:
$$L_\tau ( (\1_\em \otimes a) u (\1_\em \otimes b) ) = L_{b \cdot \tau \cdot a} (u),$$
where $u \in \em \tt \en$, $a, b \in \en$, $\tau \in \en_*$.
\end{lemma}
\begin{proof}
Due to the normality of
%the mapping $L_\tau$
%the
left slice maps, it suffices to prove the statement
%in case
for elementary tensors $u=S \otimes T \in \em \tt \en$.
% is an
%elementary tensor, say $u = x \otimes y$.
But in this case, we obtain:
\begin{eqnarray*}
L_\tau ( (\1_\em \otimes a) u (\1_\em \otimes b) ) &=& L_\tau (S \otimes aTb) \\
&=& \langle \tau, aTb \rangle ~S \\
&=& \langle b \cdot \tau \cdot a, T \rangle ~S \\
&=& L_{b \cdot \tau \cdot a} (u),
\end{eqnarray*}
%which already yields the claim.
whence the desired equality follows.
\end{proof}
The above
%module property
statement
%can
will now be used to derive a corresponding module homomorphism property shared by the
amplification of a completely bounded mapping, as announced at the beginning of this section.
%
%We now come to the announced result
\begin{satz} \label{modulhom}
Let $\em$ and $\en$ be von Neumann algebras. Then for every $\Phi \in \C\B(\em)$, the
amplified mapping $\Chi_\en (\Phi) \in \C\B(\em \tt \en)$ is an
$\1_{\em} \otimes \en$-bimodule homomorphism.
\end{satz}
\begin{proof}
As in the proof of Theorem \ref{amsatz}, we denote by $\kappa$ the pre-adjoint mapping of $\Chi_\en$. Fix
$u \in \em \tt \en$, $a, b \in \en$, $\rho \in \em_*$ and $\tau \in \en_*$.
We write $a' \dogl \1_{\em} \otimes a$, $b' \dogl \1_{\em} \otimes b$.
\par
We first note that
\begin{eqnarray}
\kappa (\rho \otimes \tau \otimes (a' u b')) =
\kappa (\rho \otimes (b \cdot \tau \cdot a) \otimes u). \label{gleichungg}
\end{eqnarray}
For
%, using
by equation (\ref{chi}), in order to establish (\ref{gleichungg}),
we only have to show that $$L_\tau (a' u b') = L_{b \cdot \tau \cdot a}(u),$$ which
%is
%true
in turn is precisely the statement of Lemma \ref{easy}.
\par
Now we
obtain for $\Phi \in \C\B(\em)$:
%see that
\begin{eqnarray*}
\langle \Chi_\en(\Phi) (a' u b'), \rho \otimes \tau \rangle &=& \langle \Phi, \kappa(\rho \otimes \tau \otimes (a' u b')) \rangle \\
&\stackrel{(\ref{gleichungg})}{=}& \langle \Phi, \kappa (\rho \otimes (b \cdot \tau \cdot a) \otimes u) \rangle \\
&\stackrel{(\ref{gleichun})}{=}& \langle \Phi, \kappa [ (b' \cdot (\rho \otimes \tau) \cdot a') \otimes u] \rangle \\
&=& \langle \Chi_\en(\Phi) (u), b' \cdot (\rho \otimes \tau) \cdot a' \rangle \\
&=& \langle a' \Chi_\en(\Phi) (u) b', \rho \otimes \tau \rangle,
\end{eqnarray*}
%whence the desired
which
%yields the claim.
finishes the proof.
\end{proof}
We
finish by
%conclude with 
remarking that Theorem \ref{modulhom} yields in particular the statement $(**)$ given above; for if
$\Phi \in \C\B(\B(\H))$, we have $\Chi_{\B(\K)}(\Phi)=\Phi^{(\infty)}$, as noted in section \ref{disc}, equation (\ref{glinf}). 
%
%\vspace{0.7cm}
%
%\begin{section}{Appendix} 
%
%\end{section} 

\vspace{0.7cm} 
\noindent
{\large{\textbf{Acknowledgments}}}
\\
%\vspace{0.02cm}
\noindent
The author is indebted to Prof.\ Dr.\ G.\ Wittstock for many
%highly valuable
very fruitful discussions on the subject of the present paper.

\begin{thebibliography}{99}

\bibitem[{\sc{deC--Haa}} 85]{haa}
{\sc{de Canni\`ere, J.; Haagerup, U.}},
{\textit{Multipliers of the Fourier algebras of some simple Lie
groups and their discrete subgroups}},
Amer.\ J.\ Math.\ 107 (1985), no.\ 2, 455--500.

\bibitem[{\sc{Cha--Smi}} 93]{chsm}
{\sc{Chatterjee, A.; Smith, R.\ R.}},
{\textit{The central Haagerup tensor product
and maps between von Neumann algebras}},
J.\ Funct.\ Anal.\ 112 (1993), no.\ 1, 97--120.

\bibitem[{\sc{Dal}} 00]{dales}
{\sc{Dales, H.G.}},
{\textit{Banach algebras and automatic continuity}}, 
London Mathematical Society Monographs New Series, 24. Clarendon Press, Oxford, 2000. 

\bibitem[{\sc{Eff--Kis}} 87]{efki}
{\sc{Effros, E.\ G.; Kishimoto, A.}},
{\textit{Module maps and Hochschild-Johnson cohomology}},
Indiana Univ.\ Math.\ J.\ (2) 36 (1987) 257--276.

\bibitem[{\sc{Eff--Lan}} 77]{efla}
{\sc{Effros, E.\ G.; Lance, C.}},
{\textit{Tensor products of operator algebras}},
Adv.\ in Math.\ 25 (1977), 1--34.

\bibitem[{\sc{Eff--Rua}} 88]{bim}
{\sc{Effros, E.\ G.; Ruan, Z.-J.}},
{\textit{Representations of operator bimodules and their applications}},
J.\ Operator Theory 19 (1988), 137--157.

\bibitem[{\sc{Eff--Rua}} 90]{efru}
{\sc{Effros, E.\ G.; Ruan, Z.-J.}},
{\textit{On approximation properties for operator spaces}},
Internat.\ J.\ Math.\ 1 (1990), no.\ 2, 163--187.

\bibitem[{\sc{Eff--Rua}} 00]{efruan}
{\sc{Effros, E.\ G.; Ruan, Z.-J.}},
{\textit{Operator spaces}}, Oxford University Press, 2000.

\bibitem[{\sc{Eff--Kra--Rua}} 93]{efkrru}
{\sc{Effros, E.\ G.; Kraus, J.; Ruan, Z.-J.}},
{\textit{On two quantized tensor products}},
Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991), 125--145, Res.\ Notes 
Math., 5, A K Peters, Wellesley MA, 1993. 

\bibitem[\textsc{Gar--Pfe} 84]{hand} 
{\sc{Gardner, R.\ J.; Pfeffer, W.\ F.}}, 
{\textit{Borel measures}}, 
in: Kunen, K.; Vaughan, J.\ E. (eds.), 
{\textit{Handbook of set-theoretic topology}}, 
North-Holland Publishing Co., Amsterdam-New York, 1984, 961--1043. 

\bibitem[{\sc{Ge--Kad}} 96]{gekad}
{\sc{Ge, L.; Kadison, R.}},
{\textit{On tensor products of von Neumann algebras}},
Invent.\ Math.\ 123 (1996), 453--466. 

\bibitem[{\sc{Hof}} 95]{hof}
{\sc{Hofmeier, H.}},
{\textit{Vollst\"andig beschr\"ankte Modul-Homomorphismen \"uber von Neumann-Algebren}},
Ph.\ D.\ Thesis, Universit\"at des Saarlandes, 1995.

\bibitem[{\sc{Hof--Wit}} 97]{wiho}
{\sc{Hofmeier, H.; Wittstock, G.}}, 
{\textit{A bicommutant theorem for completely bounded module homomorphisms}}, 
Math.\ Ann.\ 308 (1997), 141--154. 
%
%\bibitem[{\sc{Hu--Neu}} 01]{hune}
%{\sc{Hu, Zhiguo; Neufang, M.}}, 
%{\textit{Mazur's property and property $(X)$ of higher level for preduals of von Neumann algebras}}, 
%preprint. 

\bibitem[\textsc{Kan--Mag} 78]{kama} 
{\sc{Kanamori, A.; Magidor, M.}}, 
{\textit{The evolution of large cardinal axioms in set theory}}, 
Higher Set Theory, Proceedings, Oberwolfach, Germany 1977, ed.\ G.\ H.\ M\"uller and D.\ S.\ Scott, 99--275, 
Berlin-Heidelberg, New York (1978). 

\bibitem[{\sc{Kra}} 83]{krtrans}
{\sc{Kraus, J.}},
{\textit{The slice map problem for $\sigma$-weakly closed subspaces of von Neumann algebras}},
Trans.\ Amer.\ Math.\ Soc.\ 279 (1983), no.\ 1, 357--376.

\bibitem[{\sc{Kra}} 91]{kraus}
{\sc{Kraus, J.}},
{\textit{The slice map problem and approximation properties}},
J.\ Funct.\ Anal.\ 102 (1991), 116--155.

\bibitem[{\sc{May--Neu--Wit}} 89]{mnw}
{\sc{May, G.; Neuhardt, E.; Wittstock, G.}},
{\textit{The space of completely bounded module homomorphisms}},
Arch.\ Math.\ (Basel) 53 (1989), no.\ 3, 283--287.

\bibitem[{\sc{Neu}} 01]{ich1}
{\sc{Neufang, M.}}, 
{\textit{On Mazur's property and property $(X)$}}, 
preprint. 

\bibitem[{\sc{Pau}} 86]{paul}
{\sc{Paulsen, V.}},
{\textit{Completely bounded maps and dilations}},
Pitman Research Notes in Math., no.\ 146, Longman Scientific and Technical, New York, 1986.

\bibitem[{\sc{Pis}} 00]{pis}
{\sc{Pisier, G.}},
{\textit{An introduction to the theory of operator spaces}}, Notes du cours du Centre Emile Borel, 2000.

\bibitem[{\sc{Rua}} 92]{ru}
{\sc{Ruan, Z.-J.}},
{\textit{On the predual of dual algebras}},
J.\ Operator Theory 27 (1992), 179--192.
%so recht?

\bibitem[{\sc{Str}} 81]{strat}
{\sc{Str\u{a}til\u{a}, \c{S}.}},
{\textit{Modular theory in operator algebras}},
Editura Academiei, Bucure\c{s}ti, Rom\^{a}nia, Abacus Press, Tunbridge Wells, Kent, England.

\bibitem[{\sc{Str--Zsi}} 99]{stzs}
{\sc{Str\u{a}til\u{a}, \c{S}.; Zsid\'{o}, L.}},
{\textit{The commutation theorem for tensor products over von Neumann algebras}}, 
J.\ Funct.\ Anal.\ 165 (1999), 293--346. 

\bibitem[{\sc{Tom}} 70]{tom}
{\sc{Tomiyama, J.}},
{\textit{Tensor products and projections of norm one in von Neumann algebras}},
Lecture notes, University of Copenhagen, 1970.

\bibitem[{\sc{Van}} 78]{vand}
{\sc{Van Daele, A.}},
{\textit{Continuous crossed products and type III von Neumann algebras}},
London Mathematical Society Lecture Note Series, 31. Cambridge University Press, Cambridge--New York, 1978. 

\bibitem[{\sc{Wit}} \textit{et al.} 99]{witt}
{\sc{Wittstock, G.}} \textit{et al.},
{\textit{Was sind Operatorr\"aume -- Ein Internetlexikon}}, URL:
http://www.math.uni-sb.de/~ag-wittstock/Lexikon99/ (1999).


\end{thebibliography}

\vspace{0.2cm}
\noindent
{\sc{Author's address:}}
%\vspace{0.1cm}
\\
{\textit{
Department of Mathematical Sciences\\
University of Alberta\\
Edmonton, Alberta\\
Canada T6G 2G1\\
%\vspace{0.1cm}
E-mail:}}
%\begin{verbatim}
mneufang@math.ualberta.ca
%\end{verbatim}
\end{document}


math0201078.extracted_bib
\begin{thebibliography}{99}

\bibitem[{\sc{deC--Haa}} 85]{haa}
{\sc{de Canni\`ere, J.; Haagerup, U.}},
{\textit{Multipliers of the Fourier algebras of some simple Lie
groups and their discrete subgroups}},
Amer.\ J.\ Math.\ 107 (1985), no.\ 2, 455--500.

\bibitem[{\sc{Cha--Smi}} 93]{chsm}
{\sc{Chatterjee, A.; Smith, R.\ R.}},
{\textit{The central Haagerup tensor product
and maps between von Neumann algebras}},
J.\ Funct.\ Anal.\ 112 (1993), no.\ 1, 97--120.

\bibitem[{\sc{Dal}} 00]{dales}
{\sc{Dales, H.G.}},
{\textit{Banach algebras and automatic continuity}}, 
London Mathematical Society Monographs New Series, 24. Clarendon Press, Oxford, 2000. 

\bibitem[{\sc{Eff--Kis}} 87]{efki}
{\sc{Effros, E.\ G.; Kishimoto, A.}},
{\textit{Module maps and Hochschild-Johnson cohomology}},
Indiana Univ.\ Math.\ J.\ (2) 36 (1987) 257--276.

\bibitem[{\sc{Eff--Lan}} 77]{efla}
{\sc{Effros, E.\ G.; Lance, C.}},
{\textit{Tensor products of operator algebras}},
Adv.\ in Math.\ 25 (1977), 1--34.

\bibitem[{\sc{Eff--Rua}} 88]{bim}
{\sc{Effros, E.\ G.; Ruan, Z.-J.}},
{\textit{Representations of operator bimodules and their applications}},
J.\ Operator Theory 19 (1988), 137--157.

\bibitem[{\sc{Eff--Rua}} 90]{efru}
{\sc{Effros, E.\ G.; Ruan, Z.-J.}},
{\textit{On approximation properties for operator spaces}},
Internat.\ J.\ Math.\ 1 (1990), no.\ 2, 163--187.

\bibitem[{\sc{Eff--Rua}} 00]{efruan}
{\sc{Effros, E.\ G.; Ruan, Z.-J.}},
{\textit{Operator spaces}}, Oxford University Press, 2000.

\bibitem[{\sc{Eff--Kra--Rua}} 93]{efkrru}
{\sc{Effros, E.\ G.; Kraus, J.; Ruan, Z.-J.}},
{\textit{On two quantized tensor products}},
Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991), 125--145, Res.\ Notes 
Math., 5, A K Peters, Wellesley MA, 1993. 

\bibitem[\textsc{Gar--Pfe} 84]{hand} 
{\sc{Gardner, R.\ J.; Pfeffer, W.\ F.}}, 
{\textit{Borel measures}}, 
in: Kunen, K.; Vaughan, J.\ E. (eds.), 
{\textit{Handbook of set-theoretic topology}}, 
North-Holland Publishing Co., Amsterdam-New York, 1984, 961--1043. 

\bibitem[{\sc{Ge--Kad}} 96]{gekad}
{\sc{Ge, L.; Kadison, R.}},
{\textit{On tensor products of von Neumann algebras}},
Invent.\ Math.\ 123 (1996), 453--466. 

\bibitem[{\sc{Hof}} 95]{hof}
{\sc{Hofmeier, H.}},
{\textit{Vollst\"andig beschr\"ankte Modul-Homomorphismen \"uber von Neumann-Algebren}},
Ph.\ D.\ Thesis, Universit\"at des Saarlandes, 1995.

\bibitem[{\sc{Hof--Wit}} 97]{wiho}
{\sc{Hofmeier, H.; Wittstock, G.}}, 
{\textit{A bicommutant theorem for completely bounded module homomorphisms}}, 
Math.\ Ann.\ 308 (1997), 141--154. 
%
%\bibitem[{\sc{Hu--Neu}} 01]{hune}
%{\sc{Hu, Zhiguo; Neufang, M.}}, 
%{\textit{Mazur's property and property $(X)$ of higher level for preduals of von Neumann algebras}}, 
%preprint. 

\bibitem[\textsc{Kan--Mag} 78]{kama} 
{\sc{Kanamori, A.; Magidor, M.}}, 
{\textit{The evolution of large cardinal axioms in set theory}}, 
Higher Set Theory, Proceedings, Oberwolfach, Germany 1977, ed.\ G.\ H.\ M\"uller and D.\ S.\ Scott, 99--275, 
Berlin-Heidelberg, New York (1978). 

\bibitem[{\sc{Kra}} 83]{krtrans}
{\sc{Kraus, J.}},
{\textit{The slice map problem for $\sigma$-weakly closed subspaces of von Neumann algebras}},
Trans.\ Amer.\ Math.\ Soc.\ 279 (1983), no.\ 1, 357--376.

\bibitem[{\sc{Kra}} 91]{kraus}
{\sc{Kraus, J.}},
{\textit{The slice map problem and approximation properties}},
J.\ Funct.\ Anal.\ 102 (1991), 116--155.

\bibitem[{\sc{May--Neu--Wit}} 89]{mnw}
{\sc{May, G.; Neuhardt, E.; Wittstock, G.}},
{\textit{The space of completely bounded module homomorphisms}},
Arch.\ Math.\ (Basel) 53 (1989), no.\ 3, 283--287.

\bibitem[{\sc{Neu}} 01]{ich1}
{\sc{Neufang, M.}}, 
{\textit{On Mazur's property and property $(X)$}}, 
preprint. 

\bibitem[{\sc{Pau}} 86]{paul}
{\sc{Paulsen, V.}},
{\textit{Completely bounded maps and dilations}},
Pitman Research Notes in Math., no.\ 146, Longman Scientific and Technical, New York, 1986.

\bibitem[{\sc{Pis}} 00]{pis}
{\sc{Pisier, G.}},
{\textit{An introduction to the theory of operator spaces}}, Notes du cours du Centre Emile Borel, 2000.

\bibitem[{\sc{Rua}} 92]{ru}
{\sc{Ruan, Z.-J.}},
{\textit{On the predual of dual algebras}},
J.\ Operator Theory 27 (1992), 179--192.
%so recht?

\bibitem[{\sc{Str}} 81]{strat}
{\sc{Str\u{a}til\u{a}, \c{S}.}},
{\textit{Modular theory in operator algebras}},
Editura Academiei, Bucure\c{s}ti, Rom\^{a}nia, Abacus Press, Tunbridge Wells, Kent, England.

\bibitem[{\sc{Str--Zsi}} 99]{stzs}
{\sc{Str\u{a}til\u{a}, \c{S}.; Zsid\'{o}, L.}},
{\textit{The commutation theorem for tensor products over von Neumann algebras}}, 
J.\ Funct.\ Anal.\ 165 (1999), 293--346. 

\bibitem[{\sc{Tom}} 70]{tom}
{\sc{Tomiyama, J.}},
{\textit{Tensor products and projections of norm one in von Neumann algebras}},
Lecture notes, University of Copenhagen, 1970.

\bibitem[{\sc{Van}} 78]{vand}
{\sc{Van Daele, A.}},
{\textit{Continuous crossed products and type III von Neumann algebras}},
London Mathematical Society Lecture Note Series, 31. Cambridge University Press, Cambridge--New York, 1978. 

\bibitem[{\sc{Wit}} \textit{et al.} 99]{witt}
{\sc{Wittstock, G.}} \textit{et al.},
{\textit{Was sind Operatorr\"aume -- Ein Internetlexikon}}, URL:
http://www.math.uni-sb.de/~ag-wittstock/Lexikon99/ (1999).


\end{thebibliography}