A Unified Approach to the Topological Centre %Theorems Problem \\ %for $\l1^{**}$ and $\lu$
Matthias Neufang
%We show that the predual of a von Neumann algebra $\em$ enjoys Mazur's property if and Let $\G$ be a locally compact group. Consider the Banach algebra $\l1^{**}$, equipped with the first Arens multiplication, as well as the algebra $\lu$, the dual of the space of bounded left uniformly continuous functions on $\G$, whose product extends the convolution in the measure algebra $\M(\G)$. %whose product extends . We\footnote{2000 {Mathematics Subject Classification}: 22D15, 43A20, 43A22. \par {Key words and phrases}: locally compact group, group algebra, left uniformly continuous functions, Arens product, topological centre. \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.} present (for the most interesting case of a non-compact group) completely different -- in particular, direct -- proofs and even obtain sharpened versions of the results, first proved by Lau--Losert in \cite{lalo} and Lau in \cite{lau}, that the topological centres of the latter algebras precisely are $\l1$ and $\M(\G)$, respectively. The %particular special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it %is avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method %demonstrates that reveals the nature of %once more their %intimate close relation. %of both results, %can %be %obtained %%by deriving them in perfect parallelity. %%reveals the parllty of the results
math0201079.tex
\documentclass[draft,reqno,11pt,oneside]{article} 
%%%%%%%%%%%noindent's KONSEQUENT BENUTZT!?, EEOO 
%%%LUC=L1*\li schreiben, u sagen, dass hier nicht gebraucht!?, eo 

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%vorher ich: -2cm
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\begin{document} 
\newtheorem{lemma}{Lemma}[section]
\newtheorem*{eldemo}{Sketch of proof}
\newtheorem{satz}[lemma]{Theorem}
\newtheorem{prop}[lemma]{Proposition}
\newtheorem{defi}[lemma]{Definition}
\newtheorem{bei}[lemma]{Example} 
\newtheorem{verm}[lemma]{Conjecture} 
\newtheorem{kor}[lemma]{Corollary} 
\renewcommand{\proofname}{Proof} 
\newtheorem{bez}[lemma]{Notation} 
\newtheorem{bem}[lemma]{Remark}
%was hat er hier zu meckern!? kennt er nicht!? 
\newtheorem{fall}[lemma]{Case}
\newtheorem*{example}{Example} 
\newtheorem{defsatz}[lemma]{Definition and Theorem} 
%aufpassen mit bei-Umgebing von oben!! , auch mit Durchnummerierung! eeooo 
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%
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\newtheorem{satzap}[lemmaap]{Theorem}
\newtheorem{propap}[lemmaap]{Proposition}
\newtheorem{defiap}[lemmaap]{Definition}
\newtheorem{beiap}[lemmaap]{Example}  
\newtheorem{korap}[lemmaap]{Corollary}  
\newtheorem{bezap}[lemmaap]{Notation} 
\newtheorem{bemap}[lemmaap]{Remark}
%
\def \Q{{\mathbb{Q}}}
\def \Ar{{\mathcal{A}}}
\def \G{{\mathcal{G}}} 
\def \pu{{\mbox{.}}} 
\def \ps{{\Phi^s}} 
%\def \otp{{\otimes}} 
%\def \Ce{{\rm{C}}} 
\def \LUC{{\mathrm{LUC}}} 
\def \RUC{{\mathrm{RUC}(\G)}}
\def \RSp{{\widetilde{R}}^{(p)}} 
\def \RSq{{\widetilde{R}}^{(q)}} 
\def \wwp{{\mathfrak{P}}} 
\def \phii{{\overline{\Phi}}} 
\def \Mun{{\mathcal{M}}}
\def \Ce{{\mathrm{C}}}
\def \Cede{{\mathbb{C}}} 
%\def \kleinp{{\mbox{\textit{p}}}} 
\def \lpp{{L_p}} 
\def \Te{{\mathcal{T}}}
\def \sis{{\mathcal{S}}} 
\def \th{{\widetilde{\Theta}}} 
\def \mc{{\mathcal{MC}}} 
\def \ml{{\mathcal{MLUC}}} 
\def \sus{{\subseteq}} 
%oder echte Teilmenge!?????????? : Vorsicht von Datumsbeweis her!! eeooo 
\def \mr{{\mathcal{MRUC}}} 
\def \N{{\mathbb{N}}}
\def \al{{\alpha}} 
\def \a{{\alpha}} 
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\def \gurp{{\Gamma_{r, p}}} 
\def \gurrp{{{\widetilde{\Gamma_{r, p}}}}} 
%%%%%%%%%%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 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 
\def \M{{\rm{M}}}
\def \en{{\mathcal{N}}}
\def \gr{{\widetilde{\Gamma_r}}} 
%%und auch ohne Tilde def.en!!?? eo?! mit entsprechenden Eig.en eo!? 
\def \gre{{\Gamma_r}} 
\def \em{{\mathcal{M}}} 
\def \emli{{\mathcal{ML}_\infty}} 
\def \ccc{{\mathcal{C}}}
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\def \cep{{\C\B(\B(L_p(\G)))}} 
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\def \Is{{\widetilde{I}}} 
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\def \tt{{\overline{\otimes}}} 
\def \ti{{\stackrel{\vee}{\otimes}}} 
\def \oo{{{\overline{\otimes}}}} 
\def \tp{{\widehat{\otimes}}} 
\def \otp{{\otimes}} 
%echt richtig?, eo: vgl im eingebundenen tom-tex-Kap!?, eo 
\def \zet{{Z_t(\lu)}} 
%oder doch mit Z_{mathrm t} schreiben, eeeeeooooo !? 
\makeatletter 
\newcommand{\essu}
{\mathop{\operator@font ess\mbox{-}sup}} 
\newcommand{\essi}
{\mathop{\operator@font ess\mbox{-}inf}} 
\newcommand{\lisu}
{\mathop{\operator@font lim\,ess\mbox{-}sup}} 
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\def \gam{{\gamma_p}}
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\def \gur{{\Gamma_r}} 
\def \Ker{{\it{KERN}}}
\def \F{{\mathfrak{F}}}
\def \ad{{\mathrm{ad}}} 
%\def \iotas{{\widetilde{\iota}}} 
%\def \qs{{\widetilde{q}}} 
\def \Ceee{{\mathbb{C}}}
\def \Cee{{\mathrm{C}}}
\def \imink{{\iota_{\mathrm{minK}}}} 
\def \imin{{\iota_{\mathrm{min}}}} 
\def \Bild{{\it{BILD}}}
\def \Sz{{\mathcal{S}}}
%\def \ens{{{\mathcal{N}}(L_p(\G))}}
\def \The{{\mathcal{T}}}
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\def \mi{{\Gamma_2(\M(\G))}}
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\def \bsi{{\B^s(\B(\Ha))}}
\def \gu{{\widetilde{\Gamma_l}}}
\def \EF{{\widetilde{F}}} 
\def \gi{{\Gamma_l}} 
\def \als{{\widetilde{\alpha}}} 
\def \betas{{\widetilde{\beta}}} 
\def \gir{{\Gamma_r}} 
\def \muss{{\widetilde{\mu}}} 
\def \ge{{\Gamma}}
\def \gem{{\Gamma(\M(\G))}}
\def \MM{{\mathbf{M}}} 
\def \un{{\ell_{\infty}^*(\G)}}
\def \R{{\mathcal{R}}}
\def \cebr{{\C\B_{\R(\G)}(\B(L_2(\G)))}}
%\newcommand{\Chi}[1]{\left.\chi\right._{\!#1}} 
\def \Chi{{\chi}} 
\def \cebre{{\C\B_{\R(\G)}(\B(\ell_2(\G)))}}
\def \cebrs{{\C\B_{\R(\G)^{'}}^{\sigma}(\B(L_2(\G)))}}
\def \cebra{{\C\B_{\R(\G)}^{\sigma}(\B(\ell_2(\G)))}}
\def \U{{\mathfrak{U}}}
\def \K{\Ka} 
\def \cebru{{\C\B_R(\B(L_2(\G)))}}
\def \lin{{\mathrm{lin}}} 
\def \cebri{{\C\B_{R}^{\sigma}(\B(\H))}}
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\def \rem{{(e_j)^{\frac{1}{2}} \otimes M_{\widecheck{b}} \otimes (e_j)^{\frac{1}{2}}}} 
% 
%\begin{center} 
%{\Large{\textsc{A new approach to the topological center theorem for $\l1^{**}$}}}
%\end{center} 
%
%\begin{titlepage} 
\title{A Unified Approach to the Topological Centre 
%Theorems 
Problem \\ 
%for $\l1^{**}$ and $\lu$} 
for Certain Banach Algebras Arising in Abstract Harmonic Analysis} 
\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} 
%We show that the predual of a von Neumann algebra $\em$ enjoys Mazur's property if and 
Let $\G$ be a locally compact group. Consider the Banach algebra $\l1^{**}$, equipped with the 
first Arens multiplication, as well as the algebra $\lu$, the dual of the space of 
bounded left uniformly continuous functions on $\G$, whose product extends the convolution in the measure algebra $\M(\G)$. 
%whose product extends . 
We\footnote{2000 {\textit{Mathematics Subject Classification}}: 22D15, 43A20, 43A22. 
\par 
{\textit{Key words and phrases}}: locally compact group, group algebra, left uniformly continuous functions, 
Arens product, topological centre. 
\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.} 
present (for the most interesting case of a non-compact group) 
completely different -- in particular, \textit{direct} -- proofs 
and even obtain \textit{sharpened} versions of the results, first proved by Lau--Losert in 
\cite{lalo} and Lau in \cite{lau}, that the topological centres of the latter algebras precisely are 
$\l1$ and $\M(\G)$, respectively. The 
%particular 
special interest of our new approach lies in the fact that it shows a 
fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, 
it 
%is 
avoids the use of any measure theoretical techniques. At the same time, 
deriving both results in perfect parallelity, 
our method 
%demonstrates that 
reveals the nature of 
%once more 
their 
%intimate 
close relation. 
%of both results, 
%can 
%be 
%obtained 
%%by deriving them in perfect parallelity. 
%%reveals the parllty of the results
\end{abstract}

\section{Introduction} 
In this note, we wish to present a 
%new and short proof of the 
new and, for the first time, \textit{unified} approach to two theorems which 
may be considered as the fundamental results, known by now, 
concerning the topological centre problem 
%%charactizing the topcentre of/for... 
for concrete Banach algebras studied in Abstract Harmonic Analysis. 
%well-known 
Namely, for a period of nearly 15 years (beginning in the seventies with \cite{zap}), 
%the main interest 
research in the topological centre question 
was centered around the Banach algebras 
%%focused on... 
%characterization of the 
%topological centre of the Banach algebra 
$L_1(\G)^{**}$, endowed with the first Arens product, and its quotient algebra $\lu$, where $\G$ denotes a 
locally compact group. -- Here, we write $\LUC (\G)$ for the space of complex-valued bounded left uniformly continuous functions 
on $\G$; the definition of the product on the dual $\lu$ will be briefly recalled below. 
%%%besser auseinanderziehen!! erst L_1** als fernziel -- grosser-philos: \lu als zwischenstufe, dann L_1**; 
%%%eventuell auch vgl zu UCB, A(G), A_p(G), B_[rho](G), B_p, eeeooo 
%%OBEN ZAPPA, PATERSON EO erwaehnen!?, eo 
%We recall that the topological centre is defined to be the subset 
%%HIER ZAPPA & CO ERWAEHNEN, DANN: 
%%WIR Brauchen fuer lu etwas mehr!! 
The questions were eventually answered in full generality by the decisive work of Lau \cite{lau} and Lau--Losert \cite{lalo}. 
In the present note, we will derive both these results (see Theorems \ref{hau} and \ref{hau10} below) by organizing the 
arguments in a parallel fashion, 
which at the same time 
%yielding 
not only will yield direct proofs, but even sharpenings of the statements. 
%and 
\begin{satz} \label{hau}
The topological centre $Z_t(L_1(\G)^{**})$ of the Banach algebra $L_1(\G)^{**}$ is precisely $L_1(\G)$. 
\end{satz}
\begin{proof}
This 
%was first established in full generality by Lau and Losert 
is the main result, Thm.\ 1, in \cite{lalo}. 
\end{proof} 
We shall restrict ourselves to the most interesting situation where $\G$ is non-compact. In case $\G$ is compact, 
the above assertion may be attained by a very short argument (\cite{lalo}, Thm.\ 1; also cf.\ \cite{ipu}, Thm.\ 3.3 (vi)). 
\par 
The second theorem even is only of interest in the non-compact case -- for in case $\G$ is compact, 
$\lu$ equals the measure algebra $\M(\G)$, and the assertion is immediately veryfied. 
\begin{satz} \label{hau10}
The topological centre $Z_t(\lu)$ of the Banach algebra $\lu$ is $\M (\G)$. 
\end{satz}
\begin{proof}
This is the main result, Thm.\ 1, in \cite{lau}. 
\end{proof} 
%\par 
In the sequel, we will have to distinguish between the two Arens products on the bidual $\l1^{**}$. We 
denote by $\odot$ the first ($=$ left) and by $.$ the second ($=$ right) Arens product, 
and use these symbols as well for the various module operations linking $\l1$, its dual and bidual, as follows. 
Let $m , n \in \l1^{**}$, $h \in \l1^*$, $f , g \in \l1$. Denoting by $*$ the convolution product of functions 
(whenever it is defined), we write: 
\begin{center} 
\begin{tabular}{ccc} 
$\langle h \odot f, g \rangle$ & $:=$ & $\langle h, f * g \rangle$ \\ 
$\langle n \odot h, f \rangle$ & $:=$ & $\langle n, h \odot f \rangle$ \\ 
$\langle m \odot n, h \rangle$ & $:=$ & $\langle m, n \odot h \rangle$ 
\end{tabular} 
\end{center} 
%In perfekter Symmetrie erkl\"art man das rechte Arens-Produkt auf $\Ar^{**}$ gem\"a"s: 
and, following a completely symmetric pattern: 
\begin{center} 
\begin{tabular}{ccc}
$\langle f . h, g \rangle$ & $:=$ & $\langle h, g * f \rangle$ \\ 
$\langle h . m, f \rangle$ & $:=$ & $\langle m, f . h \rangle$ \\ 
$\langle m . n, h \rangle$ & $:=$ & $\langle n, h . m \rangle$. 
\end{tabular} 
\end{center} 
We note that, in particular, 
%one has explicitly 
$h \odot f = \widetilde{f} * h \in \LUC(\G)$; here, 
%and $$
%is the canonical right module operation of $\l1$ on $\li$ via convolution, 
%yielding functions in $\LUC(\G)$ 
as usual, we write $\widetilde{f} := \frac{1}{\Delta} \check{f}$, where $\Delta$ denotes the modular function of $\G$ and 
$\check{f} (x) := f(x^{-1})$ for all $x \in \G$. 
We refer to \cite{pal}, 
\S 1.4, for a discussion of basic properties of Arens multiplication in the framework of general Banach algebras. 
\par 
Whenever 
%not explicitly stated otherwise, 
we 
%shall always regard 
consider $\l1^{**}$ as a Banach algebra, we regard it as 
equipped with the first Arens product. 
We recall that the 
topological centre of $\l1^{**}$ 
is defined to be the set of functionals $m \in \l1^{**}$ which satisfy $m \odot n = m . n$ for all 
$n \in \l1^{**}$. Equivalently, the topological centre consists of all the functionals $m \in \l1^{**}$ such 
that left multiplication by $m$ is $w^*$-$w^*$-continuous on $\l1^{**}$. 
%For a discussion of the general notion 
A detailed analysis of topological centres in the general context of biduals of Banach algebras 
can be found, e.g., in \cite{laul}. 
%subset of the bidual on which both products coincide. 
\par 
%Furthermore, a
%As is usually the case, our study of $\li^*$ will to some extent involve the 
%Banach algebra 
%space $\lu$, i.e., 
%which as a Banach space is nothing but 
%the dual of the commutative $C^*$-algebra $\LUC(\G)$ of 
%complex-valued bounded left uniformly continuous functions on the group $\G$. In particular, 
There is an analogous notion of topological centre for the Banach algebra $\lu$. First, let us recall the natural 
construction of the product 
in the latter space. -- If $n \in \lu$ and $f \in \LUC(\G)$, then it is classical that 
the function $n \cdot f$, defined through $$(n \cdot f)(x) := \langle n , l_x f \rangle \quad (x \in \G)$$
still belongs to $\LUC(\G)$; i.e., $\LUC(\G)$ is left introverted. 
This operation 
%The above 
%actually 
%turns $\LUC(\G)$ into a left $\lu$-module, which 
gives rise to the 
product on the space $\lu$: 
$$\langle m \cdot n , f \rangle := \langle m , n \cdot f \rangle \quad (m , n \in \lu , f \in \LUC(G))$$ 
under which 
%making 
the latter indeed becomes a Banach algebra, and $\LUC(\G)$ becomes a left $\lu$-module with the action introduced above. 
In analogy to the case of $\l1^{**}$, one defines the topological centre of $\lu$ to be the set of 
elements $m \in \lu$ such that left multiplication by $m$ is $w^*$-$w^*$-continuous on $\lu$. 
\par 
We shall also consider the natural (left) module operation 
of $\lu$ on $\li$ given by 
$\langle m \Diamond h, g \rangle := \langle m, h \odot g \rangle$, 
where $m \in \lu$, $h \in \li$, $g \in \l1$. 
%Here, $h \odot g := \widetilde{g} * h$ is the canonical right module operation of $\l1$ on $\li$ via convolution, 
%yielding functions in $\LUC(\G)$ 
%(as usual, we write $\widetilde{g} := \frac{1}{\Delta} \check{g}$, where $\Delta$ denotes the modular function of $\G$ and 
%$\check{g} (x) = g(x^{-1})$ for all $x \in \G$). 
At this point, we note that, as is easily seen, 
one has $m \Diamond h = \widetilde{m} \odot h$, 
where $\widetilde{m}$ is an arbitrary Hahn-Banach extension of the functional $m$ to $\li^*$. 
\par 
The main interest of the approach presented here 
%is 
consists in the following: 
\begin{itemize}
\item 
We obtain a 
%stronger result
sharpening of the non-trivial inclusion in the 
%above 
statements of both Theorem \ref{hau} and \ref{hau10}. Namely, as for Theorem \ref{hau}, we shall see that for an element 
$m \in \li^*$ in order to belong to $\l1$, it is sufficient 
%that we 
to have $m \odot n = m . n$ only for all $n \in \li^*$ that are Hahn-Banach extensions of functionals in 
${\overline{\delta_\G}}^{w^*} \subseteq \Ball(\lu)$ -- instead of requiring the latter equality for all $n \in \li^*$, 
as does the definition of the topological centre. 
%In an analogous fashion, 
Analogously, in the situation of Theorem \ref{hau10}, we will prove that an element $m \in \lu$ already 
belongs to $\M(\G)$ if left multiplication by $m$ is only 
$w^*$-$w^*$-continuous on ${\overline{\delta_\G}}^{w^*} \subseteq \Ball(\lu)$, instead of demanding this 
continuity on the whole unit ball of $\lu$. 
\item 
The only proof known so far of the topological centre theorem for $\lu$ is indirect (see Thm.\ 1 in \cite{lau}), and all 
proofs given for the corresponding theorem for $\l1^{**}$ either heavily rely on the latter (\cite{laul}, \cite{ghmc}) or are also 
indirect (see \cite{lalo}, Thm.\ 1). Our 
%approach 
proofs of the two results 
%is parallel, but 
are independent, and in both cases \textit{direct}. -- We remark \textit{en passant} that the proofs of Thm.\ 5.4 and Cor.\ 5.5 
in \cite{laul} are correct only under the additional set-theoretic assumption that the compact covering number of the 
group $\G$ is a non-measurable cardinal, since the argument is based on \textit{ibid.}, Lemma 5.3, which in turn has to be read 
with a similar set-theoretic assumption; we refer to \cite{ich1} and \cite{hune} for a detailed discussion of these and related problems. 
The tool which enables us to overcome precisely those set-theoretic difficulties is provided by Lemma \ref{kmazur} below. 
\item 
The procedure we present shows a perfect analogy 
%to the proof of the corresponding result for the Banach algebra 
%$\lu$ -- stating that $Z_t(\lu)=\M(\G)$ -- which has been given in \cite{ich}, Satz 3.5.1. 
between the two topological centre theorems. Besides one additional structural result for $\lu$, the prerequisites are 
the same, and the proofs themselves follow completely parallel lines. 
%%are compl.y parallel 
\item 
The method of proof follows a purely Banach algebraic procedure and does not, in particular, rely on any 
measure theoretic argument, 
so that it might be applied 
%also 
equally well in other situations. 
%%only prerwequisites depend on actual algebra!?; sagen!?, eeoo 
At this point, we only mention that in \cite{hune}, we obtained an analogue 
of Lemma \ref{kmazur} below in the ``dual'' setting, i.e., with $\li$ replaced by the group von Neumann algebra 
$\mathit{VN} (\G)$, and with $\kapp (\G)$ replaced by $b (\G)$, the smallest cardinality of an open basis at the neutral element of $\G$. 
%Since f 
For Proposition \ref{ftcsatz} as well, 
%at least in the case of amenable groups $\G$, 
a dual version is known at least in the case of amenable groups $\G$ -- here, 
the algebras $\mathrm{UCB}(\widehat{\G})^*$ and $B_\rho (\G)$ take over the place of $\lu$ and $\M(\G)$, respectively. Thus, 
if 
%in this setting 
we are able to prove a 
%corresponding 
factorization result, for bounded families in $\mathit{VN} (\G)$ of cardinality at most 
$b (\G) \cdot \aleph_0$, corresponding to Lemma \ref{fakt}, then our method of proof 
would immediately yield an affirmative answer 
%in the affirmative 
(for the most interesting case of non-discrete groups $\G$) 
%yield that the topological centers 
%solve 
of two longstanding 
%open problems 
conjectures at once -- namely, the topological centre of the bidual of the Fourier algebra $A(\G)$ being just $A(\G)$, i.e., 
$Z_t (A(\G)^{**}) = A(\G)$, and, for amenable groups $\G$, $Z_t (\mathrm{UCB}(\widehat{\G})^*) = B_\rho (\G)$. 
%%%M(G)** erwaehnen!?? 
%%For instance, it can be 
%%proved in a completely analoguous fashion that (under certain mild assumptions) the topological centre 
%%of $\Sz_1(\G)^{**}$, when equipped with the first Arens product, is nothing but $\Sz_1(\G)$, thus 
%%revealing 
%%once more the 
%deep 
%%profound relationship of the latter to the group algebra $L_1(\G)$. Here, $\Sz_1(\G)$ denotes $\Te(L_2(\G))$, 
%%the space of trace class operators 
%%on $L_2(\G)$, when carrying 
%%the 
%Banach algebra structure 
%%multiplication which has been defined in Chapter 5.2 of \cite{ich}, rendering 
%the space $\Sz_1(\G)$ 
%%$\Te(L_2(\G))$ into 
%%a Banach algebra which can be thought of as the operator analogue of the classical convolution algebra 
%%$\l1$. In fact, as we just 
%%mentioned, we have $Z_t(\Sz_1(\G)^{**})=\Sz_1(\G)$, as is the case for $\l1$, whereas 
%%$\ell_1$ with pointwise multiplication and 
%%$\Te(L_2(\G))$ with its usual product both are Arens regular 
%%(the last assertion is a special case of \cite{dal}, Cor.\ 5.4). 
\end{itemize}

\section{Preliminaries} 
%For both proofs we only 
For both proofs 
%require 
we will use the following two lemmata, which are of interest in their own right. 
%which we both quote from \cite{ich}. 
\begin{lemma} \label{kmazur}
For an arbitrary locally compact group $\G$, the space $\l1$ enjoys Mazur's property of 
level $\kapp(\G) \cdot \aleph_0$, where $\kapp(\G)$ denotes the 
compact covering number of $\G$ 
(i.e., the least cardinality of 
a compact covering of $\G$). 
-- This means that 
a functional $m \in \l1^{**}$ actually belongs to $\l1$ if it carries 
%is $w^*$-continuous with respect to 
%all 
bounded $w^*$-converging nets of cardinality at most $\kapp(\G) \cdot \aleph_0$ into converging nets. 
\end{lemma}
\begin{proof}
This is Thm.\ 4.4 in \cite{ich1}. 
\end{proof} 
\noindent 
Next we present 
%the crucial but in fact quite general tool. 
our crucial tool, which is a general factorization theorem for 
bounded families in $\li$. It has already been used (cf.\ \cite{ich2}) 
\begin{itemize} 
\item 
to answer (in the affirmative) a question raised by Hofmeier--Wittstock in \cite{wiho} concerning the automatic 
boundedness of left $\li^*$-module homomorphisms on $\li$ 
\item 
to give an alternative approach to the result on 
automatic $w^*$-$w^*$-continuity of the latter mappings, as first shown by 
Ghahramani--McClure in \cite{ghmc}. 
\end{itemize} 
\begin{lemma} \label{fakt}
Let $\G$ be a locally compact non-compact group with compact covering number $\kapp(\G)$. Let further 
$(h_\al)_{\al \in I} \subseteq \li$ be a bounded family of functions where $|I| \leq \kapp(\G)$. Then there exist 
a family $(\psi_\al)_{\al \in I}$ of functionals in ${\overline{\delta_\G}}^{w^*} \subseteq \Ball(\lu)$ and a 
function $h \in \li$ such that the factorization formula 
$$h_\al = \psi_\al \Diamond h$$
holds for all $\al \in I$. (Moreover, the functionals $\psi_\al$, $\al \in I$, do not depend, except for the 
index set, on the given family $(h_\al)_{\al \in I}$; they are universal in the sense that they 
%only depend on the 
are obtained intrinsically from the 
group $\G$.) 
\end{lemma}
\begin{proof}
See 
\cite{ich}, Satz 3.6.2 (or \cite{ich2}). 
\end{proof} 
%%%was sagt's einem, dass mit DEM str-resultat beide beweise parallel laufen -- ueber resultate u/o ueber alg'en, 
%%%real nature, eeoo 
In order to arrange the proof of Thm.\ \ref{hau10} in a completely 
%analogous 
parallel manner, 
%The proof of Thm.\ \ref{} requires one well-known 
we only require 
%, in add., 
the following proposition, whose first part is a classical 
%and classical 
structural result about the algebra $\lu$ going back to the 
%(by now) classical. 
pioneering work of Curtis--Fig\`{a}-Talamanca (\cite{cufi}, Thm.\ 3.3). We write $\B^\sigma (L_\infty (\G))$ 
for the space of normal (i.e., $w^*$-$w^*$-continuous) operators on $L_\infty (\G)$. 
%
%%IM FOLgen ueberlegen, was man von darst wirklich braucht!?: wohl nur w*-w*-st u inj!?, eo ; letzteres aber eh das schwerste!?, eo 
%%%darst.en w*w*-st, wie meine!?, u/o andere eigschaften!?, eo 
\begin{prop} \label{ftcsatz} 
The mapping 
$$\phi : \lu \longrightarrow \B (L_{\infty}(\G))$$ 
defined through 
$$\phi (m) (h) := m \Diamond h \quad ( m \in \lu, ~h \in \li )$$ 
is an isometric representation of $\lu$ in $\B (\li)$ such that 
\begin{eqnarray} 
\phi (\M(\G)) = \phi (\lu) \cap \B^{\sigma}(\li) . \label{rel} 
\end{eqnarray} 
\end{prop}
\begin{proof}
The first assertions are well-known (see, e.g., \cite{tlau}, Thm.\ 1, together with Lemma 1 and Remark 3; or 
\cite{cufi}, Thm.\ 3.3, for a proof in the unimodular case). 
%The last 
The relation (\ref{rel}) is seen as follows (cf.\ \cite{ich}, Prop.\ 3.1.1). -- In order to prove the inclusion ``$\subseteq$'', 
let $\mu \in \M(\G)$. We have to show that $\phi (\mu) \in \B(\li)$ is normal. 
Consider a net $(h_\al)_\al \subseteq \Ball(\li)$ such that $h_\al \stackrel{w^*}{\longrightarrow} 0$. Fix $g \in \l1$. 
We claim that 
$$\langle \phi (\mu)(h_\al), g \rangle \longrightarrow 0.$$ 
But we have: 
$$\langle \phi (\mu)(h_\al), g \rangle = \langle \mu \Diamond h_\al , g \rangle = \langle \mu, {\widetilde{g}} * h_\al \rangle 
= \int_\G ({\widetilde{g}} * h_\al)(t) ~d\mu (t) ,$$ 
where (as is readily checked) 
the net 
%die Funktionenfamilie 
$({\widetilde{g}} * h_\al)_\al \subseteq \LUC(\G)$ is equicontinuous, bounded 
and converges pointwise to $0$. It thus converges uniformly on compact subsets of $\G$, 
whence we conclude that the above integrals converge to $0$, as desired. 
%gleichm\"a"sige Konvergenz auf Kompakta vor, und 
%()
%insgesamt ergibt sich die behauptete Konvergenz der obigen Integrale (bzgl.\ des Ma"ses $\mu \in \M(\G)$) gegen $0$. 
%oder wissen wir's hier schon!? 
\par 
Turning to the inclusion ``$\supseteq$'', let us consider 
an arbitrary element $\phi (m)$ of the set on the right side, where $m \in \lu$. 
%Zerlegen wir das Funktional $m \in \lu$ gem\"a"s der in Lemma \ref{lumc} angegeben 
%Bilden wir 
We denote by $\Cee_0(\G)$ the space of all complex-valued continuous functions on $\G$ 
vanishing at infinity, and by $\Cee_0(\G)^{\perp}$ its annihilator in $\lu$. Then 
we obviously have $\lu = \M(\G) \oplus \Cee_0(\G)^{\perp}$ (the latter is actually 
an $\ell_1$-direct sum -- cf.\ Lemma 1.1 in \cite{gll} --, but we will not need this fact). 
Write $m=\mu + n$ with $\mu \in \M(\G)$ and $n \in \Cee_0(\G)^{\perp}$ according to this decomposition. 
%, so ist zu 
It suffices to show that $\phi (n)=0$. 
%Dann ist zu zei
%%%%%direkt klar, ohne injtaet von psi2; auch in art so machen!?, eo 
\par 
First, using the inclusion proved above, we see that $\phi (n) = \phi (m) - \phi (\mu)$ is normal. Fix $h \in \li$. 
We have $h = \sigma(\li, \l1)-\lim_\al h_\al$ for 
an appropriate net $(h_\al)_\al \subseteq \Cee_0(\G)$. 
Hence we obtain: 
%aber: 
$$\phi (n)(h) = \sigma(\li, \l1)-\lim_\al \phi (n)(h_\al) = 0 ,$$ 
which finishes the proof. -- For the last equality, note that 
for all $\al$ and arbitrary $g \in \l1$: 
$$\langle \phi (n)(h_\al), g \rangle = \langle n, {\widetilde{g}} * h_\al \rangle = 0,$$ 
since ${\widetilde{g}} * h_\al \in L_1(\G) * \Cee_0(\G) = \Cee_0(\G)$. 
%This is part of 
%\cite{ich}, Thm.\ ??? and 
%Remark ???. 
%Prop.\ ??? (i). 
\end{proof} 
\begin{bem} 
We only stated the specific properties of $\phi$ for the sake of completeness. In the sequel, we shall just use the fact 
that $\phi$ is bounded and injective, as well as the relation $(\ref{rel})$. 
%-- note that the proof of the latter also only relies on 
%these 
%the injectivity 
%basic properties 
%of $\phi$. 
\end{bem} 
%%statt br-edw u lau(=vert.mit linksversch ist vert.mit transl---wann eig genau!?): FTC u meine norm: da ich 
%%wegen darst von lu FTC EH brauche --> kuerzer, denn sonst auch frage (=eh frage!!): 
%%klar, dass br-edw dazu past!?, eo 

\section{Application to the topological centre problem} 
We 
%now come to 
first present the proof of Theorem \ref{hau}, for non-compact groups $\G$. -- To establish the 
non-trivial inclusion, let $m \in Z_t(\l1^{**})$. The group $\G$ being non-compact, we infer from 
Lemma \ref{kmazur} that $\l1$ has Mazur's property of level $\kapp(\G)$. So to prove $m \in \l1$, let 
$(h_\al)_{\al \in I} \subseteq \li$ be a bounded net converging $w^*$ to $0$, where $|I|\leq\kapp(\G)$. 
Thanks to Lemma \ref{fakt}, we have the factorization 
$$h_\al = \psi_\al \Diamond h = 
\widetilde{\psi_\al} \odot h \quad (\al \in I)$$ 
with $\psi_\al \in {\overline{\delta_\G}}^{w^*} \subseteq \Ball(\lu)$ and $h \in \li$. -- Here, 
$\widetilde{\psi_\al}$ denotes some arbitrarily chosen Hahn-Banach extension of $\psi_\al$ to $\li^*$. 
We have to show that $a_\al := \langle m, h_\al \rangle \stackrel{\al}{\longrightarrow} 0$. Due to the 
boundedness of $(h_\al)_\al$, it suffices to prove that every convergent subnet of $(a_\al)_\al$ tends to $0$. Let 
$(\langle m, h_{\al_\beta} \rangle)_\beta$ be such a convergent subnet. Furthermore, let 
$$E := \widetilde{w^*-\lim_\gamma \psi_{\al_{\beta_\gamma}}} \in \Ball(\li^*)$$
%for some suitable convergent subnet $\widetilde{\psi_{\al_{\beta_\gamma}}}$ in $\li^*$. 
be an arbitrarily chosen Hahn-Banach extension of some $w^*$-cluster point of the net $(\psi_{\al_{\beta}})_\beta$ 
in $\Ball(\lu)$. 
\par 
We first note that 
$E \odot h=0$, since for arbitrary $g \in \l1$ we obtain: 
\begin{eqnarray*} 
\langle E \odot h, g \rangle &=& \langle E, h \odot g \rangle = 
\lim_\gamma \langle \psi_{\al_{\beta_\gamma}}, h \odot g \rangle \\ 
&=& \lim_\gamma \langle \psi_{\al_{\beta_\gamma}} \Diamond h, g \rangle 
= \lim_\gamma \langle h_{\al_{\beta_\gamma}}, g \rangle \\ 
&=& 0. 
\end{eqnarray*}
Now we conclude that (using twice the assumption $m \in Z_t(\l1^{**})$ and in particular the fact that 
$h . m \in \l1^* \odot \l1 = \LUC(\G)$, cf.\ Lemma 3.1 a) in \cite{laul}) 
\begin{eqnarray*} 
\lim_\beta \langle m, h_{\al_\beta} \rangle &=& 
\lim_\beta \langle m, \widetilde{\psi_{\al_\beta}} \odot h \rangle = 
\lim_\beta \langle m \odot \widetilde{\psi_{\al_\beta}}, h \rangle \\ 
&=& \lim_\beta \langle m . \widetilde{\psi_{\al_\beta}}, h \rangle = 
\lim_\beta \langle \widetilde{\psi_{\al_\beta}}, h . m \rangle \\ 
&=& \lim_\beta \langle \psi_{\al_{\beta}}, h . m \rangle = 
\lim_\gamma \langle \psi_{\al_{\beta_{\gamma}}}, h . m \rangle \\ 
&=& \left\langle w^*-\lim_\gamma \psi_{\al_{\beta_{\gamma}}}, h . m \right\rangle = 
\langle E, h . m \rangle \\ 
&=& \langle m . E, h \rangle = 
\langle m \odot E, h \rangle \\ 
&=& \langle m, E \odot h \rangle = 0 , 
\end{eqnarray*} 
which gives the desired convergence. 
\par 
We now turn to the proof of Theorem \ref{hau10}. 
%, where 
%in order to avoid trivialities, 
%we assume the group $\G$ to be non-compact. 
-- The inclusion $\M(\G) \subseteq Z_t (\lu)$ being 
%%fairly 
%%das mit equ \ref{rel} machen!? 
immediate and classical (cf.\ \cite{wo}, Lemma 3.1), we restrict our attention to the reverse inclusion, where 
in order to avoid trivialities, 
we 
assume the group $\G$ to be non-compact. 
%To prove 
%For the converse, in order to avoid trivialities, 
%we assume the group $\G$ to be non-compact. 
Fix $m \in Z_t (\lu)$. Then, according to Proposition \ref{ftcsatz}, we only have to prove that $\phi (m) \in \B (\li)$ is 
$w^*$-$w^*$-continuous. To this end, 
%by Lemma \ref{kmazur}, it suffices to 
%consider 
%show that if 
consider a bounded net $( h_\al )_{\al \in I} \subseteq \li$, where $|I| \leq \kapp (\G)$, 
%to avoid trivialities. 
%%u/o mit bew-umg.en!?, eo 
%is a bounded net converging 
which converges $w^*$ to $0$. 
%then $\phi (m) ( h_\al )$ converges $w^*$ to $0$. Now, 
Lemma \ref{fakt} 
yields the factorization 
$$h_\al = \psi_\al \Diamond h \quad (\al \in I)$$ 
with $\psi_\al \in {\overline{\delta_\G}}^{w^*} \subseteq \Ball(\lu)$ and $h \in \li$. 
By Lemma \ref{kmazur}, we only have to show that $\phi(m) (h_\al)$ tends $w^*$ to $0$. The latter net 
%$(h_\al)_\al$ 
being bounded, it suffices to prove that every convergent subnet 
%of $(a_\al)_\al$ 
tends to $0$. Let 
%$(\langle m, h_{\al_\beta} \rangle)_\beta$ 
$(\phi(m) (h_{{\al}_{\beta}}))_\beta$ 
be such a convergent subnet. Furthermore, let 
$$E := w^*-\lim_\gamma \psi_{\al_{\beta_\gamma}} \in {\overline{\delta_\G}}^{w^*} \subseteq \Ball(\lu)$$ 
%for some suitable convergent subnet $\widetilde{\psi_{\al_{\beta_\gamma}}}$ in $\li^*$. 
be a 
%n arbitrarily chosen Hahn-Banach extension of some 
$w^*$-cluster point of the net $(\psi_{\al_{\beta}})_\beta$ in $\Ball(\lu)$. 
\par 
We remark that 
$E \Diamond h=0$, since for arbitrary $g \in \l1$ we obtain: 
\begin{eqnarray*} 
\langle E \Diamond h, g \rangle &=& \langle E, h \odot g \rangle = 
\lim_\gamma \langle \psi_{\al_{\beta_\gamma}}, h \odot g \rangle \\ 
&=& \lim_\gamma \langle \psi_{\al_{\beta_\gamma}} \Diamond h, g \rangle 
= \lim_\gamma \langle h_{\al_{\beta_\gamma}}, g \rangle \\ 
&=& 0. 
\end{eqnarray*}
In order to conclude, we will only require the following fact concerning the compatibility 
of our various module operations, which is easy to verify (cf.\ \S 2 in \cite{lapy}): For $\psi \in \lu$, 
$h \in \li$ and $g \in \l1$, we have 
$$(\psi \Diamond h) \odot g = \psi \cdot (h \odot g) .$$ 
Now we obtain: 
\begin{eqnarray*} 
\lim_\beta \langle \phi(m)(h_{\al_{\beta}}) , g \rangle &=& \lim_\beta \langle m \Diamond h_{\al_{\beta}} , g \rangle \\ 
&=& \lim_\beta \langle m, h_{\al_{\beta}} \odot g \rangle \\ 
&=& \lim_\beta \langle m, (\psi_{\al_{\beta}} \Diamond h) \odot g \rangle \\ 
&=& \lim_\beta \langle m, \psi_{\al_{\beta}} \cdot (h \odot g) \rangle \\ 
&=& \lim_\beta \langle m \cdot \psi_{\al_{\beta}}, h \odot g \rangle \\ 
&=& \lim_\gamma \langle m \cdot \psi_{\al_{\beta_\gamma}}, h \odot g \rangle \\ 
&=& \langle m \cdot E, h \odot g \rangle \quad \mathrm{since} ~m \in Z_t(\lu) \\ 
&=& \langle m, E \cdot (h \odot g) \rangle \\ 
&=& \langle m, (E \Diamond h) \odot g \rangle \\ 
&=& 0 , 
\end{eqnarray*} 
which finishes the proof. 

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{\sc{Lau, A.\ T.-M.}}, 
{\textit{Operators which commute with convolutions on subspaces of $\li$}}, 
Colloq.\ Math.\ 39 (1978), no.\ 2, 351--359. 

\bibitem[{\sc{Lau}} 86]{lau} 
{\sc{Lau, A.\ T.-M.}}, 
{\textit{Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions 
on locally compact groups and topological semigroups}}, 
Math.\ Proc.\ Camb.\ Phil.\ Soc.\ 99 (1986), 273--283. 

\bibitem[{\sc{Lau--Los}} 88]{lalo} 
{\sc{Lau, A.\ T.-M.; Losert, V.}}, 
{\textit{On the second conjugate algebra of $L_1(\G)$ of a locally compact group}}, 
J.\ London Math.\ Soc.\ 37 (1988), 464--470. 

\bibitem[{\sc{Lau--Pym}} 90]{lapy} 
{\sc{Lau, A.\ T.-M.; Pym, J.}}, 
{\textit{Concerning the second dual of the group algebra of a locally compact group}}, 
J.\ London Math.\ Soc.\ 41 (1990), 445--460. 

\bibitem[{\sc{Lau--\"Ulg}} 96]{laul} 
{\sc{Lau, A.\ T.-M.; \"Ulger, A.}}, 
{\textit{Topological centers of certain dual algebras}}, 
Trans.\ Amer.\ Math.\ Soc.\ 348 (1996), no.\ 3, 1191--1212. 

\bibitem[{\sc{Neu}} 00]{ich}
{\sc{Neufang, M.}}, 
{\textit{Abstrakte Harmonische Analyse und Modulhomomorphismen 
\"uber von Neumann-Algebren}}, 
Ph.D.\ Thesis, Universit\"at des Saarlandes (2000). 

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

\bibitem[{\sc{Neu}} 01a]{ich2}
{\sc{Neufang, M.}}, 
{\textit{Automatic boundedness and $w^*$-continuity of module homomorphisms on 
von Neumann algebras}}, 
preprint. 
%%witt-antwort und gha-mccl, + nichtkomme varianten 
%%aber Z_t-sachen und BL_P, insbesondre S_1-Z_t-sachen und allge saetze in M(G)**-art!! 
%
%\bibitem[{\sc{Neu}} 01]{ich3}
%{\sc{Neufang, M.}}, 
%{\textit{Isometric representation of convolution algebras as completely bounded module homomorphisms 
%and a characterization of the measure algebra}}, 
%preprint. 

\bibitem[{\sc{Pal}} 94]{pal} 
{\sc{Palmer, T.\ W.}}, 
{\textit{Banach algebras and the general theory of ${}^*$-algebras. Vol.\ I. Algebras and Banach algebras}}, 
Encyclopedia of Mathematics and its Applications, 49. 
Cambridge University Press, Cambridge, 1994. 

\bibitem[{\sc{Won}} 72]{wo} 
{\sc{Wong, J.\ C.\ S.}}, 
{\textit{Invariant means on locally compact semigroups}}, 
Proc.\ Amer.\ Math.\ Soc.\ 31 (1972), no.\ 1, 39--45. 

\bibitem[{\sc{Zap}} 75]{zap} 
{\sc{Zappa, A.}}, 
{\textit{The center of the convolution algebra $C_u (\G)^*$}}, 
Rend.\ Sem.\ Mat.\ Univ.\ Padova 52 (1975), 71--83. 

\end{thebibliography}
\vspace{0.1cm} 
%\\ 
%\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}


















































math0201079.extracted_bib
\begin{thebibliography}{0}
%\bibitem[{\sc{Dal--Gha--Gr{\o}}} 98]{dal} 
%{\sc{Dales, H.\ G.; Ghahramani, F.; Gr{\o}nb{\ae}k, N.}}, 
%{\textit{Derivations into iterated duals of Banach algebras}}, 
%Studia Math.\ 128 (1998), no.\ 1, 19--54. 
%

\bibitem[{\sc{Cur--Fig}} 65]{cufi} 
{\sc{Curtis, P.\ C.; Fig\`a-Talamanca, A.}}, 
{\textit{Factorization theorems for Banach algebras}}, 
in: \textit{Function algebras}, Proceedings of an International Symposium on Function Algebras held at Tulane University, 1965 
(ed.\ F.\ T.\ Birtel), p.\ 169--185. 

\bibitem[{\sc{Gha--McC}} 92]{ghmc} 
{\sc{Ghahramani, F.; McClure, J.\ P.}}, 
{\textit{Module homomorphisms of the dual modules of convolution Banach algebras}}, 
Canad.\ Math.\ Bull.\ 35 (1992), no.\ 2, 180--185. 

\bibitem[{\sc{Gha--Lau--Los}} 90]{gll} 
{\sc{Ghahramani, F.; Lau, A.\ T.-M.; Losert, V.}}, 
{\textit{Isometric isomorphisms between Banach algebras related to locally compact groups}}, 
Trans.\ Amer.\ Math.\ Soc.\ 321 (1990), 273--283. 

\bibitem[{\sc{Hof--Wit}} 97]{wiho} 
{\sc{Hofmeier, H.; Wittstock, G.}}, 
{\textit{A bicommutant theorem for completely bounded module homomorphisms}}, 
Math.\ Ann.\ 308 (1997), no.\ 1, 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[{\sc{I\c{s}i--Pym--\"Ulg}} 87]{ipu} 
{\sc{I\c{s}ik, N.; Pym, J.; \"Ulger, A.}}, 
{\textit{The second dual of the group algebra of a compact group}}, 
J.\ London Math.\ Soc.\ 35 (1987), no.\ 2, 135--148. 

\bibitem[{\sc{Lau}} 78]{tlau} 
{\sc{Lau, A.\ T.-M.}}, 
{\textit{Operators which commute with convolutions on subspaces of $\li$}}, 
Colloq.\ Math.\ 39 (1978), no.\ 2, 351--359. 

\bibitem[{\sc{Lau}} 86]{lau} 
{\sc{Lau, A.\ T.-M.}}, 
{\textit{Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions 
on locally compact groups and topological semigroups}}, 
Math.\ Proc.\ Camb.\ Phil.\ Soc.\ 99 (1986), 273--283. 

\bibitem[{\sc{Lau--Los}} 88]{lalo} 
{\sc{Lau, A.\ T.-M.; Losert, V.}}, 
{\textit{On the second conjugate algebra of $L_1(\G)$ of a locally compact group}}, 
J.\ London Math.\ Soc.\ 37 (1988), 464--470. 

\bibitem[{\sc{Lau--Pym}} 90]{lapy} 
{\sc{Lau, A.\ T.-M.; Pym, J.}}, 
{\textit{Concerning the second dual of the group algebra of a locally compact group}}, 
J.\ London Math.\ Soc.\ 41 (1990), 445--460. 

\bibitem[{\sc{Lau--\"Ulg}} 96]{laul} 
{\sc{Lau, A.\ T.-M.; \"Ulger, A.}}, 
{\textit{Topological centers of certain dual algebras}}, 
Trans.\ Amer.\ Math.\ Soc.\ 348 (1996), no.\ 3, 1191--1212. 

\bibitem[{\sc{Neu}} 00]{ich}
{\sc{Neufang, M.}}, 
{\textit{Abstrakte Harmonische Analyse und Modulhomomorphismen 
\"uber von Neumann-Algebren}}, 
Ph.D.\ Thesis, Universit\"at des Saarlandes (2000). 

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

\bibitem[{\sc{Neu}} 01a]{ich2}
{\sc{Neufang, M.}}, 
{\textit{Automatic boundedness and $w^*$-continuity of module homomorphisms on 
von Neumann algebras}}, 
preprint. 
%%witt-antwort und gha-mccl, + nichtkomme varianten 
%%aber Z_t-sachen und BL_P, insbesondre S_1-Z_t-sachen und allge saetze in M(G)**-art!! 
%
%\bibitem[{\sc{Neu}} 01]{ich3}
%{\sc{Neufang, M.}}, 
%{\textit{Isometric representation of convolution algebras as completely bounded module homomorphisms 
%and a characterization of the measure algebra}}, 
%preprint. 

\bibitem[{\sc{Pal}} 94]{pal} 
{\sc{Palmer, T.\ W.}}, 
{\textit{Banach algebras and the general theory of ${}^*$-algebras. Vol.\ I. Algebras and Banach algebras}}, 
Encyclopedia of Mathematics and its Applications, 49. 
Cambridge University Press, Cambridge, 1994. 

\bibitem[{\sc{Won}} 72]{wo} 
{\sc{Wong, J.\ C.\ S.}}, 
{\textit{Invariant means on locally compact semigroups}}, 
Proc.\ Amer.\ Math.\ Soc.\ 31 (1972), no.\ 1, 39--45. 

\bibitem[{\sc{Zap}} 75]{zap} 
{\sc{Zappa, A.}}, 
{\textit{The center of the convolution algebra $C_u (\G)^*$}}, 
Rend.\ Sem.\ Mat.\ Univ.\ Padova 52 (1975), 71--83. 

\end{thebibliography}