On the decomposition into Discrete, type $\tw$ and type $r$ $C^*$-algebras
Chi-Keung Ng
Ngai-Ching Wong
We finish a classifying scheme of $C^*$-algebras. We show that the classes of discrete $C^*$-algebras (as defined by Peligard and Zsid\'{o}), type $\tw$, and type $r$ $C^*$-algebras (as defined by Cuntz and Pedersen) are closed under strong Morita equivalence and taking ``essential extension''. Furthermore, there exist the largest discrete finite ideal $A_{\dc,1}$, the largest discrete anti-finite ideal $A_{\dc,\infty}$, the largest type $\tw$ finite ideal $A_{\tw,1}$, the largest type $\tw$ anti-finite ideal $A_{\tw,\infty}$, and the largest type $r$ ideal $A_r$ of a $C^*$-algebra $A$ with $A_{\dc,1} + A_{\dc,\infty} + A_{\tw,1} + A_{\tw,\infty} + A_r$ being an essential ideal of $A$. When $A$ is a $W^*$-algebra, these ideals coincide with the largest type $\tI$ finite part, type $\tI$ infinite part, type $\tw$ finite part, type $\tw$ infinite part and type $r$ part, respectively. Moreover, this classification scheme observes many good rules. We find that any prime $C^*$-algebra is of one of the five types: finite discrete, anti-finite discrete, finite type $\tw$, anti-finite type $\tw$ or type $r$. If $A$ has a Hausdorff primitive spectrum, or $A$ is an $AW^*$-algebra, or $A$ is the local multiplier algebra of another $C^*$-algebra, then $A$ is a continuous field of prime $C^*$-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types. If, in addition, $A$ is discrete (respectively, anti-finite), there is an open dense subset of $\Omega$ on which each fiber is discrete (respectively, anti-finite).
1310.5464.tex
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\begin{document}

\title{On the decomposition into Discrete, type $\tw$ and type $\ttr$ $C^*$-algebras}


\author{Chi-Keung Ng  \and Ngai-Ching Wong }


\address[Chi-Keung Ng]{Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China.}
\email{ckng@nankai.edu.cn}

\address[Ngai-Ching Wong]{Department of Applied Mathematics, National Sun Yat-sen University,
Kaohsiung, 80424, Taiwan.}
\email{wong@math.nsysu.edu.tw}


\thanks{The authors are supported by National Natural Science Foundation of China (11071126), and
Taiwan NSC grant (102-2115-M-110-002-MY2).}

\date{March 3, 2014}


\keywords{$C^*$-algebra; discrete; type $\tw$; type $\ttr$; finite; anti-finite}

\subjclass[2000]{46L05, 46L35}

\begin{abstract}
We finish a classifying scheme of $C^*$-algebras.
We show that the classes of discrete $C^*$-algebras (as defined by Peligard and Zsid\'{o}),
type $\tw$,  and type $\ttr$ $C^*$-algebras (as defined by Cuntz and Pedersen)
are closed under strong Morita equivalence and taking ``essential extension''.
Furthermore, there exist the largest discrete finite ideal $A_{\dc,1}$, the largest discrete anti-finite ideal
$A_{\dc,\infty}$, the largest type $\tw$ finite ideal $A_{\tw,1}$, the largest type $\tw$ anti-finite ideal $A_{\tw,\infty}$,
and the largest type $\ttr$ ideal $A_\ttr$ of a $C^*$-algebra $A$ with $A_{\dc,1} + A_{\dc,\infty} + A_{\tw,1} + A_{\tw,\infty} + A_\ttr$ being an essential ideal of $A$.
When $A$ is a $W^*$-algebra,
these ideals coincide with the largest type
$\tI$ finite part, type $\tI$ infinite part, type $\tw$ finite part, type $\tw$ infinite part and type $\ttr$ part, respectively.
Moreover, this classification scheme observes many good rules.

We find that any prime $C^*$-algebra is of one of the five types: finite discrete, anti-finite discrete, finite type $\tw$, anti-finite type $\tw$ or type $\ttr$.
If $A$  has a Hausdorff primitive spectrum, or $A$ is an $AW^*$-algebra, or $A$ is the local multiplier algebra of another $C^*$-algebra, then $A$ is a continuous field of prime $C^*$-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types.
If, in addition, $A$ is discrete (respectively, anti-finite), there is an open dense subset of $\Omega$ on which each fiber is discrete (respectively, anti-finite).
\end{abstract}
\maketitle


\section{Introduction}

\medskip

Murray and von
Neumann
defined in \cite{Murray-vonNeumann36} (see also \cite{KRII, Murray90}) three types of  $W^*$-algebras according to the abelianness and finiteness properties of their
projections.
Since a $C^*$-algebra needs not have any non-zero projection, a similar
classification for $C^*$-algebras cannot go verbatim.
Cuntz and Pedersen defined (in \cite{CP79}) type $\tw$ and type $\ttr$ $C^*$-algebras according to certain abelianness and finiteness properties of their positive elements.
They used these, together with type $\tI$ $C^*$-algebras, to obtain a classification scheme that captures some features of the $W^*$-algebras one.

\medskip

In  \cite{NW-Mv-class}, we use open projections in $A^{**}$ to obtain another classification scheme for $C^*$-algebras parallel to the one of Murray and von Neumann.
Moreover, we observe that discrete $C^*$-algebras (as defined by Peligard and Zsid\'{o}), type $\tw$ $C^*$-algebras and type $\ttr$ $C^*$-algebras also form a good classification scheme, and some of the results in \cite{NW-Mv-class} have their counterparts in this scheme.
The original aim of this article is to presents these results.
However, we find that one has much better results and a much better classification scheme in this case.

\medskip

Let us first recall the notion of discrete, type $\tw$ and type $\ttr$ $C^*$-algebras.
An element $x\in A_+$ is said to be \emph{abelian} (in $A$) if $\overline{xAx}$ is an abelian algebra (see \cite[p.191]{Ped79}).
As in the literature, $A$ is said to be \emph{anti-liminary} if there is no non-zero abelian element in $A_+$.
Following \cite{CP79}, for $x,y\in A_+$,  we write $x\sim y$
if there is a sequence $\{z_k\}_{k\in \mathbb{N}}$ in $A$
such that $x =\sum_{k=1}^\infty z_k^*z_k$ and  $y =\sum_{k=1}^\infty z_kz_k^*$.
A positive element $x$ is said to be
\emph{finite} in $A$ if one has $y=x$ whenever $y\in A_+$ satisfying $0\leq y\leq x$ and $y\sim x$ (see, e.g., \cite{CP79}).
Notice that  a projection in a $W^*$-algebra is finite as a projection
(in the Murray-von Neumann sense) if and only if it is finite as a positive element (in the Cuntz-Pedersen sense).

\medskip

\begin{defn}[Cuntz-Pedersen; Peligard-Zsid\'{o}]
A $C^*$-algebra $A$ is said to be

\smnoind
(a) \emph{discrete}  if every non-zero element in $A_+$ dominates a non-zero abelian element in $A$ (\cite[Definition 2.1]{PZ00}).

\smnoind
(b) \emph{finite}
(respectively, \emph{semi-finite}) if every non-zero element in  $A_+$ is finite in $A$
(respectively, dominates a non-zero positive finite element  in $A$) (\cite[p.140]{CP79}).

\smnoind
(c) \emph{of type $\tw$} if it has no non-zero abelian element and it is semi-finite (\cite[p.149]{CP79}).

\smnoind
(d) \emph{of type $\ttr$} if it has no non-zero finite element (\cite[p.149]{CP79}).
\end{defn}

\medskip

A $W^*$-algebra $M$ is a type $\tI$, type $\tw$ or type $\ttr$ $W^*$-algebra
if and only if $M$ is a discrete, type $\tw$ or type $\ttr$ $C^*$-algebra (see  Proposition \ref{prop:RR0}(d)).
We also recall that a type $\tI$ $C^*$-algebra is discrete but the converse is not true (e.g.\ $\CL(\ell^2)$).
In fact, we obtained in \cite[Proposition 4.3(a)]{NW-Mv-class}
the following relation:
\begin{quotation}
\emph{A $C^*$-algebra $A$ is of type $\tI$ if and only if $A$ as well as all the primitive quotients $C^*$-algebras of $A$ are discrete.}
\end{quotation}

\medskip

It is not hard to see that $A$ is discrete if and only if every non-zero hereditary
$C^*$-subalgebra of $A$ contains a non-zero abelian hereditary $C^*$-algebra, or equivalently, a non-zero abelian element
(see \cite[Theorem 2.3]{PZ00}).
We will give alternative looks of discrete, type $\tw$ and type $\ttr$ $C^*$-algebras in a similar fashion
(see Theorem \ref{thm:sf-C-st-alg}, Remark \ref{rem:main1} and Corollary \ref{cor:type-annih-id}).
Here, by an ``annihilator ideal'', we mean the set of annihilators of another ideal.

\medskip

\begin{thm}\label{thm:main1}  Let $A$ be a $C^*$-algebra.

\smnoind
(a) $A$ is discrete if and only if every non-zero closed ideal (or equivalently, every non-zero annihilator ideal) of $A$ contains a non-zero abelian hereditary $C^*$-subalgebra.

\smnoind
(b) $A$ is of type $\tw$ if and only if there is no non-zero abelian hereditary
$C^*$-subalgebra of $A$ and every non-zero closed ideal (or equivalently, every non-zero annihilator ideal) of $A$ contains a non-zero finite hereditary $C^*$-subalgebra.

\smnoind
(c) $A$ is of type $\ttr$ if and only if it contains no non-zero finite hereditary $C^*$-subalgebra.
\end{thm}

\medskip

Moreover, we show
that types of $C^*$-algebras are invariance under strong Morita equivalence and under  essential extension.
Here, a $C^*$-algebra $B$ is an ``essential extension'' of a $C^*$-algebra $A$ if
$A$ is an essential ideal of $B$ (in the sense that $I\cap A\neq (0)$ for every non-zero closed ideal $I$ of $B$).
The following summaries  Theorem \ref{thm:sf-C-st-alg}(d), Proposition \ref{prop:type-hered},
Theorem \ref{thm:small-class} and Corollary \ref{cor:cp-type-III-anti-fin} in this paper.

\medskip

\begin{thm}
Let $A$ and $B$ be $C^*$-algebras.

\smnoind
(a) Assume that $B$ is either strongly Morita equivalent  to $A$ or
is an essential extension of  $A$.
Then  $A$ is discrete (respectively, of type
$\tw$, of type $\ttr$ or semi-finite) if and only if so is $B$.

\smnoind
(b) If $A$ is discrete (respectively, of type
$\tw$, of type $\ttr$ or semi-finite), $B$ is a hereditary $C^*$-subalgebra of $A$, and $J$ is an annihilator ideal of $A$, then both $B$ and $A/J$ are discrete (respectively, of type
$\tw$, of type $\ttr$ or semi-finite).

\smnoind
(c) The class of discrete (respectively,  semi-finite or  type $\tw$)
$C^*$-algebras is the smallest class that contains all abelian (respectively,  finite or anti-liminary finite)
$C^*$-algebras and is closed under strong Morita equivalence and
essential extension.

\smnoind
(d) $A$ is of type $\ttr$ if and only if for every hereditary $C^*$-subalgebra $B\subseteq A$, there is an essential closed ideal of $B$ with the bidual being a properly infinite $W^*$-algebra.
\end{thm}

\medskip

This gives the following result (see Corollary \ref{cor:pure-inf}).

\medskip

\begin{cor}
Suppose that $A$ is a purely infinite $C^*$-algebra (in the sense of Cuntz, Kirchberg and R\o rdam).
If $B$ is either strongly Morita equivalent to $A$ or is an essential extension of $A$ (in particular, when $B=M(A)$), then $B$ is of type $\ttr$.
\end{cor}

\medskip

Furthermore, with these new tools, we obtain the following classification scheme of
$C^*$-algebras (see  Theorems \ref{thm:inf-part} and \ref{thm:decomp}, Corollary \ref{cor:quot} and Proposition \ref{prop:decomp}).
Here, we say that a $C^*$-algebra is ``anti-finite'' if it does not contain  any non-zero
finite ideal (or equivalently, there exists an essential closed ideal with its bidual being a properly infinite $W^*$-algebra; see Proposition \ref{prop:equiv-anti-fin}(a)).

\medskip

\begin{thm}\label{thm:main3}
Let $A$ be a $C^*$-algebra.

\begin{enumerate}[a)]
\item There exists the largest discrete (respectively, semi-finite, type $\tw$ and type $\ttr$)
hereditary $C^*$-subalgebra $A_\dc$ (respectively, $A_\SF$, $A_\tw$ and $A_\ttr$) of $A$.

\item $A_\dc$, $A_\SF$, $A_\tw$ and $A_\ttr$ are ideals of $A$ such that
$A_\dc$, $A_\tw$ and $A_\ttr$ are mutually disjoint and  $A_\ttr\cap A_\SF = (0)$.

\item $A_\dc + A_\tw + A_\ttr$ is an essential   ideal of $A$, and $A_\dc + A_\tw$ is an essential   ideal of $A_\SF$.

\item $A/A_\dc$ is anti-liminary, $A/(A_\tw+A_\ttr)^{\bot\bot}$ is discrete, $A/A_\ttr$ is semi-finite, and $A/A_\SF$ is of type $\ttr$.

\item If $A$ is semi-finite, then $A/A_\tw$ is discrete and $A/A_\dc$ is of type $\tw$.

\item If $B$ is a hereditary $C^*$-subalgebra of $A$, then $B_\dc = A_\dc\cap B$, $B_\SF = A_\SF\cap B$, $B_\tw = A_\tw\cap B$ and $B_\ttr = A_\ttr\cap B$.

\item If $J$ is an essential closed ideal of $A$, then $A_\# = \{x\in A: xJ \subseteq J_\#\}$, for $\#= \dc, \SF, \tw, \ttr$.

\item If $A$ and $B$ are strongly Morita equivalent,
then $A_\#$ and $B_\#$ (for $\# = \dc, \SF, \tw, \ttr$)
corresponds to each other, under the canonical bijection between ideals of $A$
and ideals of $B$ given by an imprimitivity bimodule.

\item There exist the largest finite ideal $A_1$ and the largest anti-finite ideal $A_\infty$ of $A$.
One has $A_1\cap A_\infty = (0)$ and $A_1+ A_\infty$ is an essential   ideal of $A$.

\item $A/A_1$ is anti-finite, and  $A/A_\infty$ is a finite $C^*$-algebra.

\item If $A$ is discrete (respectively, semi-finite or of type $\tw$), then so are $A/A_1$ and $A/A_\infty$.

\item Let $J\subseteq A$ be a closed ideal, and $q:A\to A/J^\bot$ be the quotient map.
Then $J_1 = J \cap A_1$, $J_\infty = J\cap A_\infty$,
$(A/J^\bot)_1 = q(A_1)^{\bot\bot}$ and $(A/J^\bot)_\infty = q(A_\infty)^{\bot\bot}$.


\item $A_{\dc,1}:= A_\dc\cap A_1$ (respectively, $A_{\tw,1}:= A_\tw\cap A_1$) is the largest finite discrete (respectively, finite type $\tw$) ideal of $A$.
On the other hand, $A_{\dc,\infty}:= A_\dc\cap A_\infty$ (respectively, $A_{\tw,\infty}:= A_\tw\cap A_\infty$) is the largest anti-finite discrete (respectively, anti-finite type $\tw$) ideal of $A$.
One has $A_{\dc,1} + A_{\dc,\infty} + A_{\tw,1} + A_{\tw,\infty} + A_\ttr$ being an essential ideal of $A$.

\item If $A$ is a $W^*$-algebra, then $A_\dc$, $A_\tw$ and $A_\ttr$
are respectively, the type {\tI}, the type $\tw$ and the type $\ttr$ summands of $A$.
Furthermore,  $A_1$ and $A_\infty$ are respectively, the finite part and the infinite part of $A$.
\end{enumerate}
\end{thm}

\medskip

One consequence of the above is the following corollary (see Theorem \ref{thm:Glimm>prime}, Corollary \ref{cor:ess-simp} and Propositions \ref{prop:ess-simp-decomp} and \ref{prop:pure-inf>III}).

\medskip

\begin{cor}
(a) Any prime $C^*$-algebra is of one of the five types: finite discrete, anti-finite discrete, finite type $\tw$, anti-finite type $\tw$, or type $\ttr$.

\smnoind
(b) If  the primitive spectrum $\pr(A)$ of $A$ is Hausdorff, or $A$ is an $AW^*$-algebra, or $A$ is the local multiplier algebra of a $C^*$-algebra, then $A$ can be represented as the algebra of $C_0$-sections of an (F)-Banach bundle over  an open dense subset $\Omega$ of $\pr (ZM(A))$ with each fiber being a nonzero prime $C^*$-algebra of one the five types.
If, in addition, $A$ is discrete (respectively, anti-finite), there is a dense subset of  $\Omega$ on which each fiber is discrete (respectively, anti-finite).
\end{cor}

\medskip

Furthermore, for any $C^*$-algebra $A$, one has
$$A_{\dc,1} \oplus A_{\dc,\infty} \oplus A_{\tw,1} \oplus A_{\tw,\infty} \oplus A_\ttr \subseteq A \subseteq M(A_{\dc,1}) \oplus M(A_{\dc,\infty}) \oplus M(A_{\tw,1}) \oplus M(A_{\tw,\infty}) \oplus M(A_\ttr),$$
and the $C^*$-algebras $M(A_{\dc,1})$, $M(A_{\dc,\infty})$, $M(A_{\tw,1})$, $M(A_{\tw,\infty})$ and $M(A_\ttr)$ are also finite discrete, anti-finite discrete, finite type $\tw$, anti-finite type $\tw$ and type $\ttr$, respectively.

\medskip

Suppose that the spectrum of $A$ is extremely disconnected (in particular, if $A$ is $AW^*$-algebra).
Then
$$A = A_{\dc,1} \oplus A_{\dc,\infty} \oplus A_{\tw,1} \oplus A_{\tw,\infty} \oplus A_\ttr.$$
If, in addition, $ZM(A)\cong \ell^\infty(\KI)$ for some set $\KI$, then for each $i\in \KI$, there exists a  $C^*$-algebra $A_i$, which is of one of the five types, such that $\bigoplus_{i\in \KI}^{c_0} A_i \subseteq A \subseteq \prod_{i\in \KI}^{\ell^\infty} A_i$ (see Proposition \ref{prop:quasi-Ston}(b)).

\bigskip

\noindent{\it Acknowledgment}

\medskip

%changed after submission
We would like to thank Lawrence G. Brown and Joachim Cuntz for some discussion on this subject.

\medskip

\section{Notations and preliminaries}

\medskip

Throughout this article, $A$ and $B$ are $C^*$-algebras, $Z(A)$ is the center of $A$, $A_+$ is the positive cone of $A$, $\proj(A)$ is the set of non-zero (i.e. norm one) projections in $A$, $A^{**}$ is the bidual of
$A$ (equipped with the canonical $W^*$-algebra structure),
and $M(A)\subseteq A^{**}$ is the multiplier algebra of $A$.
As usual, we write $ZM(A)$ for $Z(M(A))$.
For any subsets $X,Y,Z\subseteq A$, we write $XY$ (respectively, $XYZ$)
for the linear span of $\{xy:x\in X, y\in Y\}$ (respectively, $\{xyz:x\in X, y\in Y, z\in Z\}$).
%Furthermore, we use $xYz$ %and $XyZ$
%to denote $\{x\}Y\{z\}$
%and $X\{y\}Z$ respectively,
%when $x,z\in A$.
We also write
$$X^\bot\ :=\ \{a\in A: aX = (0) = Xa\}.$$
Notice that if $X$ is an ideal of $A$, then $X^\bot$ is a closed ideal and $X^\bot = \{a\in A: aX = (0)\}$.

\medskip

A $C^*$-subalgebra  $B$  of $A$ is said to be \emph{hereditary} if $BAB\subseteq B$.
If $D\subseteq A$ is any $C^*$-subalgebra, then $D^\bot$ is a hereditary $C^*$-subalgebra.
%It is easy to see that, for every $a\in A$, the  $C^*$-subalgebra $\overline{aAa^*}$ of $A$ is  hereditary.
Moreover, if $D$ is a hereditary $C^*$-subalgebra of $B$ and $B$ is a hereditary $C^*$-subalgebra of $A$,
then $D$ is a hereditary $C^*$-subalgebra of $A$.


\medskip

Let us also give a brief account on open projections, which was introduced by Akemann in \cite{Akemann69} (see also
\cite{Akemann70,LinHX90,Ped79,Zhang89}).
A projection $p$ in $A^{**}$ is called an \emph{open projection of $A$} if
there is an increasing net $\{a_i\}_{i\in \mathfrak{I}}$ of positive elements in
$A_+$ with $\lim_i a_i = p$ in the $\sigma(A^{**},A^*)$-topology.
We use $\op(A)$ to denote the collection of non-zero open projections of $A$.
Clearly, $\proj(M(A))\subseteq \op(A)$.
In fact, $\proj(M(A))$ consists of all nonzero
open projections of $A$ which are also closed (recall that a projection $p$ in $A^{**}$ is \emph{closed} if $1-p$ is open).
%In the case when $A$ is commutative, open projections of $A$ are exactly characteristic functions of open subsets of the spectrum of $A$.

\medskip

%It is well-known that there is a canonical bijection between open (resp.\ central and open) projections and hereditary $C^*$-subalgebras (resp.\ ideals) for a $C^*$-algebra $A$.
%More precisely,
For every open (respectively, central and open) projection $e$ in $A^{**}$ the $C^*$-subalgebra
$$
\her_A(e) := eA^{**}e\cap A
$$
is hereditary (respectively, a closed ideal), and every hereditary $C^*$-subalgebra (respectively, closed ideal) of $A$ arises in this way.
The following are some well-known facts about open projections and hereditary $C^*$-subalgebras.

\medskip

\begin{prop}\label{prop:fact-open-hered}
Suppose that $A$ is a $C^*$-algebra and $e,p,q\in \op(A)$ with $p, q\in Z(A^{**})$. %(i.e.\ commutes with every elements in $A^{**}$).

\smnoind
(a) If $z(e)$ is the central cover of $e$ in $A^{**}$, then $z(e)\in \op(A)$ and $\her_A(z(e))$ is the closed ideal, $\overline{A\her_A(e)A}$, generated by $\her_A(e)$.

\smnoind
(b) $ep\in \op(A)$ and $\her_A(ep) = \her_A(e)\cap \her_A(p) = \overline{\her_A(e)\her_A(p)\her_A(e)}$.

\smnoind
(c) If $\her_A(e)\subseteq \her_A(p) +\her_A(q)$, then  $\her_A(e)\cap \her_A(p) \neq (0)$ or $\her_A(e)\cap \her_A(q) \neq (0)$.
\end{prop}

\medskip

Let us sketch a proof for part (c).
As
$
\her_A(p)+\her_A(q)\subseteq \her_A(p+q -pq),
$
we know that $e\leq p+q-pq$
and the assumption
$$
\her_A(e)\cap \her_A(p)\ =\ (0)\ =\ \her_A(e)\cap \her_A(q)
$$
will produce the contradiction that $e = e(p+q-pq) = 0$ (see part (b)).

\medskip

\begin{rem}\label{rem:essetial}
Suppose that $e,f\in \op(A)$ with $e\leq f$ and $B,D\subseteq A$ are hereditary $C^*$-subalgebras.

\smnoind
(a) There exists a largest element $r_0$ in the set
$$\{r\in \op(A)\cup \{0\}: r\leq f \text{ and } re=0\}$$
and we denote $\overline{e}^f := f-r_0$.
We say that $e$ is \emph{dense}
in $f$ if $\overline{e}^f = f$.
Notice that $\her_A(e)^\bot = \her_A(1-\overline{e}^1)$.

\smnoind
(b) $D$ is said to be \emph{disjoint} from $B$ if $BD = (0)$.
Clearly, $D$ is disjoint from $B$ if and only if $D\subseteq B^\bot$.
As in \cite{Zhang89}, we say that $B$ is \emph{essential} in $A$ if there is no non-zero hereditary $C^*$-subalgebra of $A$ that is disjoint from $B$ (or equivalently, $B^\bot = (0)$).
Thus, $e$ is dense in $f$ if and only if $\her_A(e)$ is essential in $\her_A(f)$.

\smnoind
(c) The ideal $I:=\overline{ABA}$ is essential in $A$ as a hereditary $C^*$-subalgebra if and only if for every non-zero closed ideal $J$ of $A$, one has $B\cap J \neq (0)$ (or equivalently, $B^\bot$ does not contain a non-zero ideal of $A$).
In fact, if $I$ is essential in $A$ but there is a non-zero closed ideal $J\subseteq A$ with $BJ = (0)$, then $IJ = \overline{ABA}J \subseteq \overline{ABJ} = (0)$, which is a contradiction.
Conversely, suppose that $B$ satisfies the said condition and $D\subseteq A$ is a non-zero hereditary $C^*$-subalgebra.
If $J:= \overline{ADA}$, then $BJ \neq (0)$, and hence $(0)\neq ABADA\subseteq IDA$, which implies that $ID\neq(0)$ as required.
\end{rem}

\medskip

Next, we give a brief account for the notion of strong Morita equivalence.
The readers may consult some standard literature on this subject for more information (e.g.\ \cite{Lan}).
Let $X$ be a right $A$-module equipped with an ``$A$-valued inner product'', i.e.\ a map $\la \cdot, \cdot\ra_A:X\times X \to A$
such that it is $A$-linear in the second variable, $\la x, y\ra_A^* = \la y, x\ra_A$, $\la x,x \ra_A \geq 0$ and $\la x,x \ra_A = 0$
exactly when $x=0$.
If $X$ is complete under the norm defined by $\|z\|:= \|\la z, z\ra_A\|^{1/2}$ ($z\in X$), then it is called a \emph{Hilbert $A$-module}.
A Hilbert $A$-module is said to be \emph{full} if the linear span of $\{\la x, y\ra_A: x,y\in X\}$ is dense in $A$.
For any $x,y\in X$, we define $\theta_{x,y}: X\to X$ by $\theta_{x,y} z := x\la y,z\ra_A$ ($z\in X$).
The closure, $\CK_A(X)$, of the linear span of $\{\theta_{x,y}:x,y\in X\}$ in the Banach space of bounded linear operators on $X$ is naturally a $C^*$-algebra with involution $\theta_{x,y}^* := \theta_{y,x}$.
If $B$ is another $C^*$-algebra such that there exists a full Hilbert $A$-module $X$ with $B\cong \CK_A(X)$,
then we say that $A$ and $B$ are \emph{strongly Morita equivalent}.

\begin{rem}\label{rem:str-Mori-equiv}
Let $X$ be a full Hilbert $A$-module and $B= \CK_A(X)$.

\smnoind
(a) One may equip the conjugate vector space $\ti X$ of $X$ with a canonical Hilbert $B$-module structure: $y\ti\ b = (b^*y)\ti\ $
%$\ti y\theta_{u,v} = (v\la u,y\ra)\ti\ $ \
and $\la x\ti\ , y\ti\ \ra_B := \theta_{x,y}$ ($x, y\in X; b\in B$).
As $X$ is full, for every $a\in A$, the map defined by $\Psi(a)(x\ti\ ) := (xa^*)\ti\ $
belongs to $\CK_B(\ti X)$, and $\Psi$ is an isomorphism from $A$ onto $\CK_B(\ti X)$.

\smnoind
(b) One may equip $D:=\Big(\begin{array}{cc}
A & \tilde X\\
X & B
\end{array}\Big)$ with a canonical $C^*$-algebra structure.
%such that
%$$\Big(\begin{array}{cc}
%a & x\ti\ \\
%y & b
%\end{array}\Big)^*\ :=\ \Big(\begin{array}{cc}
%a^* & y\ti\ \\
%x & b^*
%\end{array}\Big)$$
%and
%$$\Big(\begin{array}{cc}
%a & x\ti\ \\
%y & b
%\end{array}\Big)\Big(\begin{array}{cc}
%c & u\ti\ \\
%v & d
%\end{array}\Big)
%\ :=\ \Big(\begin{array}{cc}
%ac + \la x,v\ra_A & \Psi(a)(u\ti\ ) + x\ti\ d\\
%yc + bv & \theta_{y, u} + bd
%\end{array}\Big).$$
Consider $e:= \Big(\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\Big)\in M(D)$.
Then
$A \cong eDe  = \her_D(e)$ and
$B \cong (1-e)D(1-e) = \her_D(1-e)$.
Moreover, as the closed ideal of $D$ generated by $A$ is
the whole algebra $D$ and the same is true for $B$, we know from Proposition \ref{prop:fact-open-hered}(a) that $z(e) = 1 = z(1-e)$.
As a more explicit reference, the readers may consult, e.g.\ \cite[Theorem II.7.6.9]{Blac}.

\smnoind
(c) For every closed ideal $I$ of $A$, we set $\overline{XI}$  to be closed linear span of $\{xa: x\in X;a\in I\}$.
Then $\overline{XI}$ is a full Hilbert $I$-module, and
$\CK_I(\overline{XI})$ is a closed ideal of $B$.
Moreover, the map $I\mapsto \CK_I(\overline{XI})$ is a bijection from the set of closed ideals of $A$ onto those of $B$.

\smnoind
(d) Suppose that $C$ is a hereditary $C^*$-subalgebra of $A$ such that $\overline{ACA} = A$.
Then $C$ is strongly Morita equivalent to $A$.
In fact, one may take $X := \overline{CA}$ and equipped it with the canonical Hilbert $A$-module structure.
\end{rem}

\medskip

\section{Two news looks of discreteness, type $\tw$ and type $\ttr$}

\medskip

Let us start with the following simple lemma, which is implicitly included in \cite{CP79}.

\medskip

\begin{lem}\label{lem:finite}
Let $B\subseteq A$ be a non-zero hereditary $C^*$-subalgebra and $\RF^A$ be the set of all non-zero positive elements that are finite in $A$.

\smnoind
(a) If $b\in A$ satisfying $bb^*,b^*b\in B$, then $b\in B$.

\smnoind
(b) $\RF^B = \RF^A\cap B$.

\smnoind
(c) If $x\in \RF^A$ and $0<\epsilon <\|x\|$, then $\overline{(x-\epsilon)_+A(x-\epsilon)_+}$ is a finite $C^*$-algebra.

\smnoind
(d) If $x\in \RF^A$ such that $0$ is an isolated point of $\sigma(x)\cup \{0\}$ (in particular, if $x$ is a projection), then $\overline{xAx}$ is a finite $C^*$-algebra.
\end{lem}

\medskip

The next lemma is a crucial step toward the new looks and the classification scheme.


\medskip

\begin{lem}\label{lem:morita-equiv}
Let $A$ and $B$ be two strongly Morita equivalent $C^*$-algebras.
Then $A$ has a non-zero abelian element  (respectively, finite elements) if and only if $B$ does.
\end{lem}
\begin{prf}
We will only give the proof for the statement concerning finite elements since the other statement
follows from a similar argument.
Let $D$ and $e\in \proj(M(D))\subseteq \op(D)$ be as in Remark \ref{rem:str-Mori-equiv}(b).
In particular, $A = \her_D(e)$ and $z(e) = 1$ in $\op(D)$.

Suppose that $\RF^B\neq \emptyset$.
Lemma \ref{lem:finite}(b) implies that $\RF^D\neq \emptyset$.
By Lemma \ref{lem:finite}(c), we know that $D$ contains a non-zero finite hereditary $C^*$-subalgebra $D_0$.
Let $p\in \op(D)$ with $D_0 = \her_D(p)$.
Then \cite[Theorem 1.9]{PZ00} gives $e_0,e_1,p_0,p_1\in \op(D)\cup \{0\}$ such that $e_0,e_1\leq e$, $p_0, p_1\leq p$, $e_0e_1=0$, $p_0p_1=0$,
$$\overline{e_0 + e_1}^e =e, \ \ \overline{p_0 +
p_1}^p =p, \ \ z(e_0)z(p_0) = 0 \ \text{ and }\ \her_D(e_1)\cong \her_D(p_1)$$
(notice that the equivalence relation in \cite{PZ00} produces a $^*$-isomorphism between $\her_D(e_1)$ and $\her_D(p_1)$).

Let us first show that $p_1\neq 0$.
Suppose on the contrary that $p_1 =0$.
Then $e_1 = 0$ and $z(e_0)$ is
dense in $z(e) = 1$ (by \cite[Lemma 1.8]{PZ00}).
This implies that
$z(p_0) = 0$ and hence $p_0=0$, which is absurd because $p_0$ is dense in the non-zero open projection $p$.

Now, $\her_D(p_1)\subseteq D_0$ is a non-zero finite hereditary $C^*$-subalgebra of $D$.
Since $\her_D(e_1)\cong \her_D(p_1)$, we conclude that $\her_D(e_1)$ is a non-zero finite hereditary $C^*$-subalgebra
of $\her_D(e)$.
Consequently, $\RF^A \neq \emptyset$ (by Lemma \ref{lem:finite}(b)) as required.
\end{prf}

\medskip

\subsection{The first set of new looks}

\medskip


Note that there is no ambiguity in part (b) of the following theorem because of Lemma \ref{lem:finite}(b).

\medskip

\begin{thm}\label{thm:sf-C-st-alg}
(a) $A$ is discrete if and only if every non-zero closed ideal of $A$ contains a non-zero abelian element.

\smnoind
(b) $A$ is semi-finite if and only if every non-zero closed ideal of $A$ contains a non-zero finite element.

\smnoind
(c) $A$ is of type $\tw$ if and only if it is anti-liminary and every non-zero closed ideal of $A$ contains a non-zero finite element.

\smnoind
(d) Let $A$ and $B$ be two strongly Morita equivalent $C^*$-algebras.
If $A$ is discrete (respectively, anti-liminary, semi-finite, of type $\tw$ or of type $\ttr$), then so is $B$.
\end{thm}
\begin{prf}
(a) It is clear that if $A$ is discrete, the required condition holds.
Conversely, suppose that $A$ satisfies the condition as stated, and $D\subseteq A$ is a non-zero hereditary $C^*$-algebra.
If $I := \overline{ADA}$, then Remark \ref{rem:str-Mori-equiv}(d) tells us that $I$ is strongly Morita equivalent to $D$.
Now, we know from Lemma \ref{lem:morita-equiv} that $D$ contains a non-zero abelian element and hence $A$ is discrete (by \cite[Theorem 2.3]{PZ00}).

\smnoind
(b) Since the ``only if part'' is clear, we only need to consider the ``if part''.
Let $a\in A_+\setminus \{0\}$ and set $D:= \overline{aAa}$.
By Remark \ref{rem:str-Mori-equiv}(d), $D$ is strongly Morita equivalent to $\overline{ADA}$.
The hypothesis and Lemma \ref{lem:morita-equiv} produces $x\in \RF^D = \RF^A\cap D$ with $\|x\|=1$.
Notice that if $a^{1/2}x^{1/2} = 0$, we have a contradiction that $Dx^{1/2} = 0$.
Moreover, since $x^{1/2}ax^{1/2} \leq \|a\| x$ and
$a^{1/2}xa^{1/2} \sim x^{1/2}ax^{1/2}$, we know from \cite[Lemma 4.1]{CP79} that $a^{1/2}xa^{1/2}\in \RF^A$.
As $a^{1/2}xa^{1/2}\leq a$, we see that $A$
is semi-finite.

\smnoind
(c) This follows from part (b).

\smnoind
(d) This follows directly from parts (a), (b) and (c) as well as Lemma \ref{lem:morita-equiv}.
%\smnoind
%(e) Suppose that $B$ is of type $\tw$.
%If $A$ contains a non-zero abelian element, then Lemma \ref{lem:morita-equiv} will give a non-zero abelian element of $B$, which is impossible.
%Let $I$ be a non-zero closed ideal of $A$ and $I(B)$ be as in Remark \ref{rem:str-Mori-equiv}(d).
%Then $I(B)$ is strongly Morita equivalent to $I$.
%Thus, part (b) and Lemma \ref{lem:morita-equiv} tells us that $I$ will contain  a non-zero finite element, and so $A$ is of type $\tw$ (again, by part (b)).
%The arguments for the other three situations (i.e.\ discrete, type $\ttr$ and semi-finite) are similar.
\end{prf}

\medskip


\begin{rem}\label{rem:main1}
(a) It is clear that one can replace ``abelian element'' in Theorem \ref{thm:sf-C-st-alg}(a) by ``abelian hereditary $C^*$-subalgebra''.
Moreover, by Lemma \ref{lem:finite}(c), one may also replace ``finite element'' in Theorem \ref{thm:sf-C-st-alg}(b)\&(c) by ``finite hereditary $C^*$-subalgebra''.
The same reason also tells us  that:
\begin{quotation}
a $C^*$-algebra is of type $\ttr$ if and only if it has no non-zero finite hereditary $C^*$-subalgebra.
\end{quotation}

\smnoind
(b) Remark \ref{rem:str-Mori-equiv}(d) and Theorem \ref{thm:sf-C-st-alg}(d) implies that every discrete (respectively, anti-liminary, semi-finite, type $\tw$ or type $\ttr$) hereditary $C^*$-subalgebra of $A$ is contained in a discrete (respectively, anti-liminary, semi-finite, type $\tw$ or type $\ttr$) closed ideal of $A$.
\end{rem}

\medskip

\subsection{Hereditary property}

\medskip

With the help of Theorem \ref{thm:sf-C-st-alg}, we will obtain
in Proposition \ref{prop:type-hered} below some hereditary properties of discreteness,
type $\tw$, type $\ttr$ and semi-finiteness.
Apart from part (a), these properties seem to be new.
Notice that, by Remark \ref{rem:essetial}(c), the hypothesis  in part (b)
means that $B^\bot$ does not contain any non-zero ideal of $A$, but the presentation in the statement here seems more informative.

\medskip

\begin{prop}\label{prop:type-hered}
Let $B$ be a discrete (respectively, type
$\tw$, type $\ttr$, semi-finite or anti-liminary) $C^*$-algebra.

\smnoind
(a) If $A$ is a hereditary $C^*$-subalgebra of $B$, then $A$ is discrete (respectively, of type $\tw$, of type $\ttr$, semi-finite or anti-liminary).

\smnoind
(b) If $B$ is a hereditary $C^*$-subalgebra of a $C^*$-algebra $A$ such that
$I:=\overline{ABA}$ is an essential ideal of $A$, then $A$ is discrete (respectively, of type
$\tw$, of type $\ttr$, semi-finite or anti-liminary).
In particular, $M(B)$ is discrete (respectively, of type $\tw$, of type
$\ttr$, semi-finite or anti-liminary).

\smnoind
(c) If $B$ is a closed ideal of a $C^*$-algebra $A$, then both $B^{\bot\bot}$ and $A/B^\bot$ are discrete
(respectively, of type $\tw$, of type $\ttr$, semi-finite or anti-liminary).

\smnoind
(d) If $I$ is an ideal of $B$, then $B/I^\bot$ is discrete
(respectively, of type $\tw$, of type $\ttr$, semi-finite or anti-liminary).
\end{prop}
\begin{prf}
(a) This part follows from the definitions.

\smnoind
(b) We will only consider the case when $B$ is of type $\tw$ (as the argument for the other cases are similar).
By Remark \ref{rem:str-Mori-equiv}(d) and Theorem \ref{thm:sf-C-st-alg}(d), we know that $I$ is of type $\tw$.
Suppose that $(0)\neq D\subseteq A$ is a hereditary $C^*$-subalgebra.
Then $ID$ is non-zero.
As $\overline{DID}\neq (0)$ and it is a hereditary $C^*$-subalgebra of $I$, we know that  $D$ contains a non-zero finite element.
On the other hand, if $D = \overline{aAa}$ for an abelian element $a\in A_+$, then $\overline{DID}$
is a non-zero abelian hereditary $C^*$-subalgebra  of $I$, which is impossible.
These show that $A$ is of type $\tw$.

\smnoind
(c) Let $\varphi: A\to M(B)$ be the canonical $^*$-homomorphism.
As $\ker\varphi = B^\bot$, we know that $\varphi|_{B^{\bot\bot}}$ is injective and so is the induced $^*$-homomorphism $\hat \varphi:A/B^\bot \to M(B)$.
Since both $\varphi(B^{\bot\bot})$ and $\hat \varphi(A/B^\bot)$ contains the image of $B$ as an essential ideal, the conclusion follows from part (b).

\smnoind
(d) This part follows directly from parts (a) and (c).
\end{prf}


\medskip

\begin{defn}\label{defn:annih-id}
(a) We say that $A$ is an \emph{essential extension} of $B$ if it contains $B$ as an essential ideal.

\smnoind
(b) An ideal $I\subseteq A$ is called an \emph{annihilator ideal} if $I = J^\bot$ for some ideal $J\subseteq A$.
\end{defn}

\medskip

Proposition \ref{prop:type-hered}(b)  implies that the property of being
discrete, type $\tw$, type $\ttr$, semi-finite and anti-liminary are stable under essential extensions.
Furthermore, Proposition \ref{prop:type-hered}(d) states that these types are preserved under quotient by annihilator ideals.
In contrast, the quotient of a discrete $C^*$-algebra by an arbitrary closed ideal needs not be discrete (e.g.\ $\CL(\ell^2)/\CK(\ell^2)$ is of type $\ttr$).
We also note that, if $B$ is a $W^*$-algebra, then an ideal of $B$ is an annihilator ideal if and only if it is $\sigma(B, B_*)$-closed.
Thus, the phenomena in Proposition \ref{prop:type-hered}(d) extends the corresponding situation for $W^*$-algebras.

\medskip

Suppose that $I\subseteq A$ is an essential closed ideal of $A$ and $(0)\neq D\subseteq A$ is a hereditary $C^*$-algebra.
Then $I \cap D = \overline{DID}\neq (0)$.
On the other hand, a closed ideal $J\subseteq A$ is always an essential ideal of $J^{\bot \bot}$.
These, together with Remark \ref{rem:main1}, gives the following.

\medskip

\begin{cor}\label{cor:type-annih-id}
$A$ is discrete (respectively, semi-finite) if and only if every non-zero annihilator ideal of $A$ contains a non-zero abelian (respectively, finite) hereditary $C^*$-subalgebra.
\end{cor}


\medskip

\subsection{The second set of new looks}

\medskip

The following can also be regarded as another new looks of discreteness, semi-finiteness and type $\tw$.

\medskip

\begin{thm}\label{thm:small-class}
Let $\CA$ (respectively, $\CF$ and $\CF_{al}$) be the class of all abelian (respectively, finite and anti-liminary finite) $C^*$-algebras and $\D$ (respectively, $\CS$ and $\CT$) be the class obtained from $\CA$ (respectively, $\CF$ and $\CF_{al}$) by taking strong Morita equivalences and essential extensions.
Then $\D$ (respectively, $\CS$ and $\CT$) is the class of discrete (respectively, semi-finite and type $\tw$) $C^*$-algebras.
\end{thm}
\begin{prf}
By Proposition \ref{prop:type-hered},
%(which actually implies Theorem \ref{thm:decomp}(e)),
we know that every member in $\D$ is discrete.
Conversely, if $A$ is a non-zero discrete $C^*$-algebra, \cite[Theorem 2.3(iii)]{PZ00} gives an abelian hereditary $C^*$-subalgebra $A_1$ of $A$ such that $\overline{AA_1A}$ is an essential ideal.
Thus, by Remark \ref{rem:str-Mori-equiv}(d) and Proposition \ref{prop:type-hered}(b), we know that $A\in \D$.

Secondly, as in the above, every member in $\CS$ is semi-finite.
For the other inclusion, it suffices to
show that for every semi-finite $C^*$-algebra $A$, there is a finite hereditary $C^*$-subalgebra $A_2\subseteq A$
such that $\overline{AA_2A}$ is an essential ideal of $A$.
%By Lemma \ref{lem:finite}(c), $\CF_A$ is non-empty.
Indeed, by Zorn's Lemma, there exists a maximal family $\{B_i\}_{i\in \KI}$  of  finite hereditary  $C^*$-subalgebras of $A$
such that $\overline{AB_iA}$ and $\overline{AB_jA}$ are disjoint for all $i\neq j$ in $\KI$, and we set
$$
A_2\ :=\ \overline{{\sum}_{i\in \KI} B_i}.
$$
Because $A_2$ is the $c_0$-direct sum of finite $C^*$-algebras, it is finite.
Moreover, $\overline{AA_2A}$ is essential, by the maximality of $\{B_i\}_{i\in \KI}$ and Remark \ref{rem:main1}(a).

The statement concerning $\CT$ follows from the same argument as in the second case (note that the $c_0$-direct sum of anti-liminary $C^*$-algebras is anti-liminary).
\end{prf}

\medskip

\subsection{$W^*$-algebras and $C^*$-algebras of real rank zero}

\medskip

Next, we want to compare discreteness, type $\tw$ and type $\ttr$ with the corresponding properties of $W^*$-algebras.
These comparisons could be known, but since we do not find them in the literature, we present them in Proposition \ref{prop:RR0} below for later reference.

\medskip

Note that this proposition follows from Lemma \ref{lem:finite}(c),
the fact that all hereditary $C^*$-subalgebras of a real rank zero $C^*$-algebra are of real rank zero, and the fact that for any $W^*$-algebra $M$ and $p\in \proj(M)$, $pMp$ is finite as a $W^*$-algebra if and only if it is finite as a $C^*$-algebra (this fact follows from \cite[Theorem 3.4]{CP79}
and the corresponding fact for finite $W^*$-algebras).

\medskip

\begin{prop}\label{prop:RR0}
Let $A$ be a $C^*$-algebra with real rank zero.

\smnoind
(a) $A$ is discrete (respectively, semi-finite) if and only if every non-zero projection in $A$ dominates a non-zero projection which is an abelian element (respectively, finite in $A$).

\smnoind
(b) $A$ is of type $\tw$ if and only if every non-zero projection in $A$ is not abelian but dominates a
non-zero projection which is finite in $A$.

\smnoind
(c) $A$ is of type $\ttr$ if and only if it has no non-zero projection which is finite in $A$.

\smnoind
(d) If, in addition, $A$ is a $W^*$-algebra, then $A$ is
a type {\tI}, type $\tw$, type $\ttr$ or semi-finite $W^*$-algebra
if and only if it is, respectively, a discrete, type $\tw$, type $\ttr$ or semi-finite $C^*$-algebra.
\end{prop}

\medskip

\subsection{Purely infinite $C^*$-algebras and tracially infinite $C^*$-algebras}

Another application of Proposition \ref{prop:type-hered} is the following proposition, which shows that type $\ttr$ is weaker than the notion of purely infinite $C^*$-algebras 
as defined by Pedersen and  Cuntz \cite{CP79} (in the case of simple $C^*$-algebras) as well as by Kirchberg and R\o rdam \cite{KR00} (in the general case).
Note that Proposition \ref{prop:pure-inf>III} implies, in particular, \cite[Proposition 4.4]{KR00}.

\medskip

In fact, we find another property that looks very similar to type $\ttr$ (compare Proposition \ref{prop:pure-inf>III}(a) and Corollary \ref{cor:cp-type-III-anti-fin}) and lies between type $\ttr$ and pure infinity. 
This property will also be needed in Section 6 below. 
Before introducing this property, let us first consider traces on $C^*$-algebras.
In the following, we denote by $\T(A)$ the set of all lower semi-continuous semi-finite traces on $A$,
and by $T_s(A)$ the set of tracial states on $A$.
Here, a trace $\tau$ of $A$ is semi-finite if
$\tau(x) = \sup\{\tau(y): 0\leq y\leq x \text{ and } \tau(y)<+\infty\}$
for every $x\in A_+$
(see \cite[Definition 6.1.1]{Dix77}).
Note that this semi-finiteness is slightly stronger than the one in \cite{CP79}.
%Clearly, the constant null function $0$ belongs to $\T(A)$.

\medskip


\begin{propdefn}\label{prop:pure-inf>III}
$A$ is said to be \emph{tracially infinite} if it contains an essential closed ideal $J$ with $\T(J) = \{0\}$.

\smnoind
(a) $A$ is tracially infinite if and only if $A$ contains an essential ideal $J$ such that for every hereditary $C^*$-subalgebra $B\subseteq J$, the $W^*$-algebra $B^{**}$ is properly infinite.

\smnoind
(b) If $A$ is tracially infinite, then $A$ is of type $\ttr$.
In particular, every purely infinite $C^*$-algebra is of type $\ttr$.
\end{propdefn}
\begin{prf}
(a) It suffices to show that $\T(A) = \{0\}$ if and only if $T_s(B) = \emptyset$ for every non-zero hereditary $C^*$-subalgebra $B\subseteq A$.

Suppose on the contrary that $\T(A) = \{0\}$ but there exist a non-zero hereditary $C^*$-subalgebra $B\subseteq A$ and an element $\varphi\in T_s(B)$.
Let $\tau$ be the lower semi-continuous trace extension of $\varphi$ as given in the proof of \cite[Lemma 4.6]{CP79}; namely,
$$
\tau(x):= \sup \{\varphi(y): y\in B_+, y\sim z\leq x\}.
$$
It is easy to see that $\tau$ is semi-finite, in the sense of \cite[Definition 6.1.1]{Dix77}, which gives a contradiction.

Conversely, if $\tau\in \T(A)\setminus \{0\}$, then the semi-finiteness of $\tau$ will produce an element $x\in A_+$ with $0 < \tau(x) < \infty$.
Without loss of generality, assume that $\|x\| = 1$.
It is clear that $\|x - (x-\frac{1}{n})_+\| \to 0$ and the lower semi-continuity of $\tau$ tells us that $\tau\big((x-\frac{1}{n})_+\big) \to \tau(x)$.
Set $y:=(x-\frac{1}{n_0})_+$, where $n_0$ is big enough so that $\tau\big((x-\frac{1}{n_0})_+\big) > 0$.
As in the argument of \cite[Lemma 4.1]{CP79}, there is $g\in C(\sigma(x))_+$ and $\lambda >0$ such that
$y = yg^{1/2}(x)$ and $g(x) \leq \lambda x$.
Thus,
$$\tau(z)\ =\ \tau\big(g^{1/2}(x)zg^{1/2}(x)\big)\ \leq\ \lambda \|z\| \tau(x)\ <\ \infty \qquad (z\in (\overline{yAy})_+),$$
and $T_s(\overline{yAy}) \neq \emptyset$.


\smnoind
(b) Let $J$ be an essential closed ideal of $A$ with $\T(J) = \{0\}$.
By Proposition \ref{prop:type-hered}(b), it suffices to show that $J$ is of type $\ttr$.
Suppose on the contrary that $J$ contains a non-zero finite hereditary $C^*$-subalgebra $B$.
Then \cite[Theorem 3.4]{CP79} implies that $T_s(B)\neq \emptyset$, which is a contradiction (see the argument of part (a)).
Now, if $A$ is purely infinite, it follows from \cite[Proposition 5.1]{KR00} that $\T(A) = \{0\}$ and $A$ is tracially infinite.
\end{prf}

\medskip

Proposition \ref{prop:pure-inf>III} tells us that tracial infinity stands between pure infinity and type $\ttr$.
Note that pure infinity passes to quotients (see \cite[Theorem 4.19]{KR00}) but the quotient of a type $\ttr$ $C^*$-algebra can be semi-finite (see \cite[Remark 3.14]{CP79}).
On the other hand, we do not know whether tracial infinity coincides with type $\ttr$.
One can find in \cite{Ng-ext-lscsf-tr} some equivalences of the statement: ``every type $\ttr$ $C^*$-algebra is tracially infinite''.

\medskip

The following is a direct application of Proposition \ref{prop:pure-inf>III}, Theorem \ref{thm:sf-C-st-alg}(d) and Proposition \ref{prop:type-hered}.

\medskip

\begin{cor}\label{cor:pure-inf}
Let $A$ be a purely infinite $C^*$-algebra.

\smnoind
(a) If $A$ is strongly Morita equivalent to a $C^*$-algebra $B$, then $B$ is of type $\ttr$.

\smnoind
(b) An essential extension of $A$ (in particular, $M(A)$) is of type $\ttr$.
\end{cor}

\medskip

Notice that if $A$ and $B$ are separable $C^*$-algebras that are strongly Morita equivalent with $A$ being purely infinite, then \cite[Theorem 4.23]{KR00} implies that $B$ is also purely infinite.
The same may be true for non-separable $C^*$-algebras but we do not find it in the literature.

\medskip

\begin{eg}\label{eg:III-not-pi}
For every non-unital $C^*$-algebra $B$, the $C^*$-algebras $M(B\otimes \mathcal{O}_\infty)$ and $M(B\otimes \mathcal{O}_2)$ are of type $\ttr$ (by Corollary \ref{cor:pure-inf}(b) and \cite[Proposition 4.5]{KR00}).
\end{eg}

\medskip

\section{Anti-finite $C^*$-algebras and an equivalent form of type $\ttr$ $C^*$-algebras}

\medskip

As seen in Theorem \ref{thm:small-class}, stability under both strong Morita equivalence as well as essential extension plays an important role in understanding discrete and type $\tw$ $C^*$-algebras.
Clearly, finiteness is not preserved under strong Morita equivalence (e.g.\ $\mathbb{C}$ is strongly Morita equivalent to $\CK(\ell^2)$).
However, it is stable under essential extension, as stated in part (a) of the following lemma.
This lemma could be known, but we present it here for completeness.


\medskip

\begin{lem}\label{lem:finite-under-extension}
(a) If $B$ is a finite $C^*$-algebra, then every essential extension of $B$ is finite.

\smnoind
(b) If $I$ is a non-zero closed ideal of $A$, every $\tau\in T_s(I)$ extends to an element of $T_s(A)$.
\end{lem}
\begin{prf}
(a) Since a  $C^*$-subalgebra of a finite $C^*$-algebra is finite, we need only to consider the essential extension $M(B)$ of $B$.
By \cite[Theorem 3.4]{CP79}, $T_s(B)$ separates points of $B_+$.
For any element $\tau\in T_s(B)$, let $(\pi_\tau, \KH_\tau, \xi_\tau)$ be its GNS construction.
If $\pi_0:= \bigoplus_{\tau\in T_s(B)} \pi_\tau$, then $\pi_0$ is faithful and hence extends to a faithful $^*$-representation $\ti \pi_0$ of $M(B)$.
On the other hand, each $\sigma\in T_s(B)$ extends to a tracial state $\bar \sigma$ on $D := \bigoplus_{\tau\in T_s(B)}^{\ell^\infty} \pi_\tau(B)''$ such that
$$\bar \sigma\big((x_\tau)_{\tau\in T_s(B)}\big)
\ =\ \langle \pi_\sigma(x_\sigma) \xi_\sigma, \xi_\sigma\rangle  \qquad \big((x_\tau)_{\tau\in T_s(B)}\in D\big).$$
As $\ti \pi_0(M(B))\subseteq D$,  we can define $\ti \sigma := \bar \sigma\circ \ti \pi_0$.
It is easy to see that $(\pi_{\ti\sigma}, \KH_{\ti \sigma})$ is the canonical extension of $(\pi_\sigma, \KH_\sigma)$ on $M(B)$.
Moreover, we have $\ker \pi_{\ti \sigma}=\{x\in M(B): \ti \sigma(x^*x) = 0\}$, since $\ti \sigma$ is a tracial state.
Now, if $a\in M(B)_+$ satisfying $\ti \sigma(a) = 0$ for every $\sigma\in T_s(B)$, then $\pi_{\ti \sigma}(a^{1/2}) = 0$ ($\sigma\in T_s(B)$), which gives $\ti \pi_0(a^{1/2}) = 0$ and hence $a^{1/2} = 0$.
Consequently, $M(B)$ is finite (by \cite[Theorem 3.4]{CP79}).

\smnoind
(b) As $I$ is an essential closed ideal of $I^{\bot\bot}$, the argument of part (a) tells us that $\tau$ can be extended to a tracial state $\tau_0$ on $I^{\bot\bot}$.
Furthermore, as $I^{\bot\bot}+I^\bot$ is an essential ideal of $A$,
the trivial extension of $\tau_0$ on $I^{\bot\bot}+I^\bot$ (i.e.\ it vanishes on $I^\bot$) will again extends to a tracial state on $A$.
\end{prf}

\medskip

Motivated by the classification theory of  $W^*$-algebras, we make the following definition of anti-finiteness.
Note that a $W^*$-algebra is anti-finite if and only if it is properly infinite, but the name ``properly infinite $C^*$-algebras'' is already in used.

\medskip

\begin{defn}
A $C^*$-algebra $A$ is said to be \emph{anti-finite} if it contains no non-zero finite closed ideal.
\end{defn}

\medskip


We set
$$A_\infty\ := \ \bigcap \{J^\bot: J \text{ is a finite ideal of }A\} \qquad \text{and} \qquad A_1:=A_\infty^\bot.$$
It is easy to see that $A_\infty$ is the largest anti-finite ideal of $A$.
We will show that $A_1$ and $A_\infty$ can be regarded  as the ``finite part'' and the ``infinite part'' of $A$, respectively.
Let us first give the following lemma.


\medskip

\begin{lem}\label{lem:inf-part}
Let $\CJ$ be a maximal collection of mutually disjoint non-zero finite ideals of $A$.
Then $J_0:= \overline{\sum_{J\in \CJ} J}$ is a finite ideal of $A$ such that $J_0+ A_\infty$ is essential in $A$.
Moreover, $A_\infty = J_0^\bot$.
\end{lem}
\begin{prf}
Obviously, $J_0$ is finite.
Suppose that $I$ is a non-zero closed ideal of $A$.
If $I$ contains a non-zero finite ideal, then $I\cap J_0\neq (0)$ (otherwise, $\CJ$ cannot be a maximal family).
If $I$ does not contain a non-zero finite ideal, then $I\subseteq A_\infty$.
This shows that $J_0+A_\infty$ is essential.

Secondly, it is clear that $A_\infty \subseteq J_0^\bot$.
Moreover, as $J_0^\bot$ cannot contain any non-zero finite ideal (because $\CJ$ is maximal), $J_0^\bot$ is anti-finite and is contained in $A_\infty$.
\end{prf}

\medskip

Observe that the construction in Lemma \ref{lem:inf-part}
needs not give a maximal finite ideal of $A$.  For example, if $A=\ell^\infty$,
then $J_0$ could be $c_0$ with a suitable choice of a maximal collection of disjoint finite ideals,
while $\ell^\infty$ is itself finite.
%However, we have the following result.

\medskip

\begin{thm}\label{thm:inf-part}
Let $A$ be a $C^*$-algebra and $I \subseteq A$ be a closed ideal of $A$.

\smnoind
(a) Both $A_1$ and $A/A_\infty$ are finite $C^*$-algebras.

\smnoind
(b) $A/A_1$ is anti-finite and $A_\infty = A_1^\bot$.

\smnoind
(c) If $A$ is discrete (respectively, of type $\tw$, semi-finite or anti-liminary), then so are $A/A_1$ and $A/A_\infty$.

\smnoind
(d) $I_1 = I \cap A_1$ and $I_\infty = I \cap A_\infty$.

\smnoind
(e) $A_1/(I^\bot)_1$ and $A_\infty/(I^\bot)_\infty$ are essential ideals of $(A/I^\bot)_1$  and $(A/I^\bot)_\infty$ respectively.

\smnoind
(f) If $A$ is a $W^*$-algebra, then $A_1$ and $A_\infty$ are respectively the finite part and the infinite part of $A$.
\end{thm}
\begin{prf}
(a) Let $\CJ$ and $J_0$ be as in Lemma \ref{lem:inf-part}.
Suppose that $\varphi: A\to M(J_0)$ is the canonical $^*$-homomorphism.
Since $\varphi|_{A_1}$ is a $^*$-monomorphism (because $\ker \varphi = A_\infty$), we know that $A_1$ is finite (by Lemma \ref{lem:finite-under-extension}(a)).
The finiteness of $A/A_\infty$ follows from a similar argument.

\smnoind
(b) Suppose that $\theta:A \to M(A_\infty)$ is the canonical embedding.
Then $\ker\theta= A_1$ and $\theta$ induces a $^*$-monomorphism $\hat \theta:A/A_1\to M(A_\infty)$.
Since $\hat \theta(A/A_1)$ contains $\theta(A_\infty)$ as an essential ideal, if
$A/A_1$ contains a non-zero finite ideal $J$, then $\hat\theta(J)\cap \theta(A_\infty)$ will be a non-zero finite ideal of $\theta(A_\infty)$ which is a contradiction.
Thus, $A/A_1$ is anti-finite.
By a similar  argument, we know that $A_1^\bot$ is anti-finite and hence is contained in $A_\infty$.
Consequently, $A_1^\bot = A_\infty$.

\smnoind
(c) This part follows from part (b), the definition of $A_1$ as well as Proposition \ref{prop:type-hered}(c).

\smnoind
(d) If $J$ is any closed ideal of $A$, by considering an approximate unit of $I$, one has
\begin{equation}\label{eqt:orth-comp}
J^\bot \cap I = (J\cap I)^\bot \cap I.
\end{equation}
This gives
$$A_\infty \cap I
\ =\  \bigcap \{(J\cap I)^\bot\cap I: J \text{ is a finite ideal of }A\}\\
%& = & \bigcap \{K^\bot\cap I: K \text{ is a finite ideal of }I\}
\ =\ I_\infty,$$
and $A_1\cap I = A_\infty^\bot \cap I = (A_\infty \cap I)^\bot \cap I = I_\infty^\bot \cap I= I_1$.

\smnoind
(e) Let $q:A\to A/I^\bot$ be the quotient map.
By part (d) and Relation \eqref{eqt:orth-comp}, one has
$$(I^\bot)_1 = I^\bot \cap A_1 = (I_1)^\bot \cap A_1$$
and the canonical $^*$-homomorphism from $A_1$ to $M(I_1)$ induces an injection from $q(A_1) = A_1/(I^\bot)_1$ to $M(I_1)$.
Hence, Lemma \ref{lem:finite-under-extension}(a) tells us that $q(A_1)$ is finite and $q(A_1)\subseteq q(A)_1$.
Now, suppose that $J$ is a non-zero closed ideal of $q(A)_1$.
Since $q$ restricts to an injection from $q^{-1}(J)\cap I^{\bot\bot}$ to the finite ideal $J$, we know that $q^{-1}(J)\cap I^{\bot\bot}\subseteq A_1$ and is non-zero (because $q^{-1}(J)\nsubseteq I^\bot$).
Thus, $J\cap q(A_1)\neq (0)$, and $q(A_1)$ is an essential ideal of $q(A)_1$.

Similar arguments as above show that $q(A_\infty)$ can be regarded as an essential extension of $I_\infty$ (and hence $q(A_\infty)\subseteq q(A)_\infty$) and that $q(A_\infty)$ is an essential ideal of $q(A)_\infty$.


\smnoind
(f) Since $J^\bot$ is $\sigma(A,A_*)$-closed for every ideal $J\subseteq A$, we see that $A_1$ is the largest $\sigma(A,A_*)$-closed finite ideal of $A$.
This gives the required conclusion.
\end{prf}

\medskip

\cite[Theorem 3.4]{CP79} tells us that $A$ is finite if and only if $\bigcap_{\tau \in T_s(A)} \ker \pi_\tau = (0)$ (we use the convention that this intersection is $A$ when $T_s(A) = \emptyset$).
One also has the following description of anti-finite $C^*$-algebras in terms of tracial states.

\medskip

\begin{prop}\label{prop:equiv-anti-fin}
(a) The following statements are equivalent.
\begin{enumerate}
\item $A$ is anti-finite
\item $\bigcap_{\tau\in T_s(A)} \ker \pi_\tau$ is an essential ideal of $A$.
\item There is an essential ideal $J$ of $A$ with $J^{**}$ being a properly infinite $W^*$-algebra.
\end{enumerate}

\smnoind
(b) If $A$ is separable, then $A$ is anti-finite if and only if  $\ker \pi_\tau$ is an essential ideal of $A$ for every $\tau \in T_s(A)$
\end{prop}
\begin{prf}
(a) Set $J := \bigcap_{\tau\in T_s(A)} \ker \pi_\tau$.

\smnoind
$1) \Rightarrow 2)$.
Suppose on the contrary that $J^\bot \neq (0)$.
Since the representation  $\bigoplus_{\tau\in T_s(A)} \pi_\tau$ is injective on $J^\bot$, one knows that $T_s(A)$ separates positive elements of $J^\bot$.
Thus, we obtain the contradiction that $J^\bot$ is a finite closed ideal of $A$.

\smnoind
$2) \Rightarrow 3)$.
Assume that there is $\tau\in T_s(J)$.
By Lemma \ref{lem:finite-under-extension}(b), $\tau$ extends to an element $\tau_0\in T_s(A)$.
This gives the contradiction that $\tau(J_+) =\tau_0(J_+) = \{0\}$.
Hence, $T_s(J)= \emptyset$, and the bidual of the essential ideal $J$ is properly infinite.

\smnoind
$3) \Rightarrow 1)$.
Suppose on the contrary that $A$ contains a non-zero finite closed ideal $I$.
Then $I_0:= I \cap J\neq (0)$.
As $I_0$ is finite, it has a tracial state $\tau$ and $\tau$ extends to a tracial state on $J$ (by Lemma \ref{lem:finite-under-extension}(b)), which is impossible.

\smnoind
(b) Assume that there is a non-zero finite closed ideal $I\subseteq A$.
By \cite[Corollary 3.6]{CP79}, there is a faithful tracial state $\tau$ on $I$ and it extends to $\tau_0\in T_s(A)$.
It is easy to see that $I \cap \ker \pi_{\tau_0} = (0)$ and hence $\ker \pi_{\tau_0}$ is not essential.
The converse follows directly from part (a).
\end{prf}

\medskip

The above and Remark \ref{rem:main1}(a) give the following.
The reader may note the similarity and different between Statement (3) below and Proposition \ref{prop:pure-inf>III}(a).

\medskip

\begin{cor}\label{cor:cp-type-III-anti-fin}
The following statements are equivalent.
\begin{enumerate}
\item $A$ is of type $\ttr$.
\item All hereditary $C^*$-subalgebras of $A$ are anti-finite.
\item For every hereditary $C^*$-subalgebra $B\subseteq A$, there is an essential ideal $I$ of $B$ with $I^{**}$ being a properly infinite $W^*$-algebra.
\end{enumerate}
\end{cor}

\medskip

Proposition \ref{prop:equiv-anti-fin}(a) and the main result of \cite{Ng-str-amen-rep} also give the following.

\medskip

\begin{eg}
If $G$ is a separable connected non-amenable locally compact group, then
the reduced group $C^*$-algebra $C^*_r(G)$ is anti-finite.
\end{eg}

%\medskip

%We end this section with the following characterization of type $\ttr$ that is similar to Proposition \ref{prop:equiv-anti-fin}(a).
%As in \cite{CP79}, we denote by $\T(A)$ the set of lower semi-continuous semi-finite trace on $A$.
%The following lemma follows from the argument of \cite[Lemma 4.6]{CP79}.

%\medskip

%\begin{lem}
%If $J$ is any ideal of $A$, then for any $\varphi\in \T(J)$,
%$\varphi_0(a) := \sup \{\varphi(y): y\in J, \varphi(y) < \infty \text{ and } y\leq b  \text{ for some } b\sim a\}$
%defines an element in $\T(A)$. Q: is $\varphi_0$ semi-finite?
%\end{lem}


%\medskip

%\begin{prop}\label{prop:equiv-ttr}
%$A$ is type $\ttr$ if and only if there is an eseential ideal $I$ of $A$ such that for any non-zero hereditary $C^*$-subalgebra $B\subseteq I$, one has $T_s(B) = \emptyset$.
%\end{prop}
%\begin{prf}
%$\Rightarrow)$.
%For each $\varphi\in \T(A)$, the set $L_\varphi:= \{a\in A: \varphi(a^*a) = 0\}$ is a closed ideal of $A$.
%We put $I:= \bigcap_{\varphi\in \T(A)} L_\varphi$.
%Suppose on the contrary that $I^\bot \neq (0)$ and take any $x\in I^\bot_+\setminus \{0\}$.
%If $y\in I^\bot_+$ such that $y\leq x$ and $y\sim x$, then


%\end{prf}

\medskip

\section{The classification scheme in terms of discreteness, type $\tw$ and type $\ttr$}

\medskip

In this section, we will consider the remaining statements in Theorem \ref{thm:main3}.

\medskip

\begin{lem}\label{lem:ideal}
(a) If $A$ is not discrete, then $A$ contains  a
non-zero closed ideal of  either type $\tw$ or type $\ttr$.

\smnoind
(b) Let $I$ and $J$ be closed ideals of $A$ such that $I\cap J = (0)$ and $I+J$ is essential.
If $\varphi:A\to M(I)$ is the canonical $^*$-homomorphism, then $\varphi$ restricts to an injection on $J^\bot$.
\end{lem}
\begin{prf}
(a) By Theorem \ref{thm:sf-C-st-alg}(a), there is a non-zero anti-liminary closed ideal $I\subseteq A$.
If $I$ does not contain a non-zero finite element as well, then it is of type $\ttr$.
Otherwise, $I$ will contain a non-zero finite hereditary $C^*$-subalgebra $B$.
The non-zero ideal $J := \overline{IBI}$ of $I$ is semi-finite because of Remark \ref{rem:str-Mori-equiv}(d)
and Theorem \ref{thm:sf-C-st-alg}(d).
Hence, $J$ is of type $\tw$.

\smnoind
(b) If $J^\bot \cap I^\bot\neq (0)$, then $J^\bot \cap I^\bot \cap (I+J) \neq (0)$,
and Proposition \ref{prop:fact-open-hered}(c) gives the contradiction that either $J^\bot \cap I^\bot \cap I\neq (0)$ or $J^\bot \cap I^\bot \cap J \neq (0)$.
Consequently, $\varphi|_{J^\bot}$ is injective (as $\ker \varphi = I^\bot$).
\end{prf}

\medskip

\begin{thm}\label{thm:decomp}
Let $A$ be a $C^*$-algebra.

\smnoind
(a) There exists the largest discrete (respectively, semi-finite, type $\tw$ and type $\ttr$) hereditary $C^*$-subalgebra $A_\dc$ (respectively, $A_\SF$, $A_\tw$ and $A_\ttr$) of $A$.
Moreover, $A_\dc$, $A_\SF$, $A_\tw$ and $A_\ttr$ are ideals of $A$ such that $A_\dc$, $A_\tw$ and $A_\ttr$ are mutually disjoint and $A_\ttr\cap A_\SF = (0)$.

\smnoind
(b) $A_\dc + A_\tw + A_\ttr$ is an essential ideal of $A$ and $A_\dc + A_\tw$ is an essential ideal of $A_\SF$.

\smnoind
(c) $A_\al:=A_\dc^\bot$ is the largest anti-liminary hereditary $C^*$-subalgebra of $A$.

\smnoind
(d) $A_\dc = A_\al^\bot = (A_\tw + A_\ttr)^\bot$, $A_\tw = A_\SF\cap A_\dc^\bot$, $A_\dc = A_\SF\cap A_\tw^\bot$, $A_\ttr = A_\SF^\bot =  (A_\dc + A_\tw)^\bot$ and $A_\SF=A_\ttr^\bot$.

\smnoind
(e) If $A$ is a $W^*$-algebra, then $A_\dc$, $A_\tw$ and $A_\ttr$ are respectively, the type {\tI}, the type $\tw$ and the type $\ttr$ summands of $A$.
\end{thm}
\begin{prf}
(a) We first show the existence of the largest type $\tw$ hereditary $C^*$-subalgebra and that it is an ideal.
Let $\CJ_\tw$ be the collection of all type $\tw$ closed ideals of $A$.
Suppose that one has $J_1, J_2 \in \CJ_\tw$.
If $J_1+J_2$ contains a non-zero
hereditary $C^*$-algebra $B$, then by Proposition
\ref{prop:fact-open-hered}(c), either $B\cap J_1\neq (0)$ or $B\cap J_2\neq (0)$.
If $B\cap J_1\neq (0)$, there is $x\in \RF^{B\cap J_1}\subseteq \RF^B$.
On the other hand, if $B = \overline{a(J_1+J_2)a}$ for a non-zero
abelian element $a \in (J_1+J_2)_+$, then $B\cap J_1\neq (0)$ or $B\cap J_2\neq (0)$,
but they are both abelian, which contradicts $J_1, J_2\in \CJ_\tw$.
Thus, $J_1 + J_2\in \CJ_\tw$, and $\CJ_\tw$ is a directed set.

For any nonzero closed ideal $J$ of $A$, we denote by $p_J$ the element in  $\op(A)\cap Z(A^{**})$ with $J = \her_A(p_J)$.
Put
$$A_\tw\ :=\ \overline{{\sum}_{J\in
\CJ_\tw} J}.$$
Then $p_{A_\tw} = w^*$-$\lim_{J\in \CJ_\tw}
p_J$.
Suppose that $e\in \op(A)$  satisfying $\her_A(e)\subseteq
A_\tw$.
Then $$
e\ =\ e p_{A_\tw} e\ =\ w^*\text{-}{\lim}_{J\in \CJ_\tw} e p_J e,
$$
and there is $J\in \CJ_\tw$ with $ep_Je \neq 0$, which means that
$\her_A(e)\cap J \neq (0)$.
Hence, $\her_A(e)\cap J$ contains a
non-zero finite element.
On the other hand, if $\her_A(e) = \overline{aAa}$ for a non-zero abelian element $a\in A_+$, then $\her_A(e)\cap J$ will contain a non-zero abelian element, which contradicts $J$ being type $\tw$.
Consequently, one has $A_\tw\in \CJ_\tw$.
Finally, if $B\subseteq A$
is a hereditary $C^*$-subalgebra of type $\tw$, then, by Remark \ref{rem:str-Mori-equiv}(d) and Theorem \ref{thm:sf-C-st-alg}(d), one knows that $B\subseteq \overline{ABA}\subseteq A_\tw$.

The existences of $A_\SF$, $A_\tw$ and $A_\ttr$ as well as the fact that they are ideals follow from similar arguments.
The disjointness statements are obvious.

\smnoind (b)
This part follows directly from Lemma
\ref{lem:ideal}(a) (every non-zero non-discrete closed ideal interests either  $A_\tw$ or $A_\ttr$).

\smnoind
(c) If $A_\al$ contains a non-zero abelian element $x$, then $\overline{xAx}\subseteq A_\dc$ and we have a contradiction that $x\in A_\al\cap A_\dc$.
Thus, $A_\al$ is anti-liminary.
On the other hand, if $B\subseteq A$ is a non-zero anti-liminary hereditary $C^*$-subalgebra, then $\overline{BA_\dc B} = A_\dc \cap B = (0)$, which means that $B\subseteq A_d^\bot = A_\al$.

\smnoind
(d) Obviously, $A_\dc\subseteq A_\al^\bot \subseteq (A_\tw+A_\ttr)^\bot$.
As $A_\dc\cap (A_\tw+A_\ttr) = (0)$ and $A_\dc + A_\tw + A_\ttr$ is
an essential ideal of $A$, Lemma \ref{lem:ideal}(b) gives a $^*$-monomorphism from
$(A_\tw+A_\ttr)^\bot$ to $M(A_\dc)$ whose image clearly contains the image of $A_\dc$.
Therefore, Proposition \ref{prop:type-hered}(b) tells us that $(A_\tw+A_\ttr)^\bot$ is discrete.
Consequently, $(A_\tw+A_\ttr)^\bot\subseteq A_\dc$.

Secondly, as $A_\tw A_\dc  = (0)$, one knows that $A_\tw \subseteq A_\SF\cap A_\dc^\bot$.
Moreover, the argument as above produces a $^*$-monomorphism from $A_\SF\cap A_\dc^\bot$ to $M(A_\tw)$ and implies $A_\SF\cap A_\dc^\bot\subseteq A_\tw$.
The equality $A_\dc = A_\SF\cap A_\tw^\bot$ follows from exactly the same argument.

Next, we set $J_3:= (A_\dc + A_\tw)^\bot$.
As $A_\ttr A_\SF  = (0)$ and $A_\dc + A_\tw\subseteq A_\SF$,  we see that $A_\ttr \subseteq A_\SF^\bot \subseteq J_3$.
In order to establish the equalities concerning $A_\ttr$, it remains to show that $J_3$ is of type $\ttr$.
In fact, if $J_3$ has a non-zero finite hereditary $C^*$-subalgebra $B$, since $J_3\subseteq A_\dc^\bot$, one has $B\cap A_d = (0)$ and hence $B$ is of type $\tw$, which contradicts $J_3\subseteq A_\tw^\bot$.

The proof for the last equality is basically the same as that for $A_\dc = A_\al^\bot$.

\smnoind
(e) This follows from Proposition \ref{prop:RR0}(d) and the fact that $A_\dc$, $A_\tw$ and $A_\ttr$ are $\sigma(A,A_*)$-closed (because of part (d)).
\end{prf}

\medskip

This, together with Theorem \ref{thm:sf-C-st-alg}(d), produces the following result concerning the stability of the above decomposition under strong Morita equivalence (see Remark \ref{rem:str-Mori-equiv}(c) for the notation).

\medskip

\begin{cor}
If $X$ is a full Hilbert $A$-module and $B:= \CK_A(X)$, then
$B_\dc = \CK_{A_\dc}(\overline{XA_\dc})$, $B_\tw= \CK_{A_\tw}(\overline{XA_\tw})$ and $B_\ttr= \CK_{A_\ttr}(\overline{XA_\ttr})$.
\end{cor}

\medskip

On the other hand, Theorem \ref{thm:decomp} and Proposition \ref{prop:type-hered}(c) also give the following.

\medskip

\begin{cor}\label{cor:quot}
(a) $A/A_\dc$ is anti-liminary and $A/A_\al$ is discrete.

\smnoind
(b) $A/A_\ttr$ is semi-finite and $A/A_\SF$ is of type $\ttr$.

\smnoind
(c) $A_\SF/A_\tw$ is discrete and $A_\SF/A_\dc$ is of type $\tw$.
\end{cor}

\medskip

Furthermore, we want to see how type decompositions pass to hereditary $C^*$-subalgebras, essential extensions and quotients.

\medskip

\begin{prop}\label{prop:decomp}
Let $B\subseteq A$ be a hereditary $C^*$-subalgebra, $I, J\subseteq A$ be closed ideals with $J$ being essential, and $Q: A \to A/I^\bot$ be the quotient map.

\smnoind
(a) $B_\dc = A_\dc\cap B$, $B_\SF = A_\SF\cap B$, $B_\al = A_\al\cap B$,  $B_\tw = A_\tw\cap B$ and $B_\ttr = A_\ttr\cap B$.

\smnoind
(b) $(A/I^\bot)_\# = Q(\{a\in A: aI\subseteq I_\#\})$ and $Q(A_\#)$ is an essential ideal of $(A/I^\bot)_\#$, for $\#= \dc, \SF, \al, \tw, \ttr$.
In particular, $A_\# = \{a\in A:aJ \subseteq J_\#\}$.

\smnoind
(c) $A_\tw = \{x\in A: xJ_\dc = (0) \text{ and } xJ \subseteq J_\SF\}$

\smnoind
(d) $A_\ttr = \{x\in A: xJ_\SF = (0)\} = \{x\in A: xJ_\dc = (0)  \text{ and } xJ_\tw = (0)\}$.
\end{prop}
\begin{prf}
(a) This part follows from the definitions.

\smnoind
(b) First of all, observe that the statement concerning $Q(A_\#)$ being an essential ideal follows from a similar argument as that of Theorem \ref{thm:inf-part}(e).

Secondly, we show that $A_\#$ coincides with the closed ideal
$$\check A_\#\ :=\ \{x\in A: xJ \subseteq J_\#\}$$
for $\# = \dc, \SF, \al, \tw, \ttr$.
In fact, we will only consider the case when $\# = \tw$ and the other cases follows from similar arguments.
Since $A_\tw J \subseteq A_\tw\cap J = J_\tw$ (by part (a)), we have $A_\tw \subseteq \check A_\tw$.
Let $B\subseteq \check A_\tw$ be any non-zero hereditary $C^*$-subalgebra.
Then $B\cap J = \overline{BJB}$ is contained in $J_\tw$.
As $J$ is essential, one knows that $BJ\neq (0)$ and there exists a non-zero finite element in $B\cap J$.
On the other hand, if $B = \overline{aA a}$ for a non-zero abelian element $a$, then $B\cap J$ is a non-zero abelian hereditary $C^*$-subalgebra of $J_\tw$, which is a contradiction.
Consequently, $\check A_\tw$ is of type $\tw$ and is contained in $A_\tw$.

Next, we consider the equality concerning $I_\#$.
Note that, by considering the canonical $^*$-monomorphism $\varphi:A/I^\bot\to M(I)$, one may regard $A/I^\bot$ as an essential extension of $I$.
Now, the statement concerning $J_\#$ implies the asserted relation, since we have
\begin{equation*}
\varphi((A/I^\bot)_\#)
\ = \ \{\varphi(x): x\in A/I^\bot; \varphi(x)I \subseteq I_\#\}
\ = \ \{\varphi(Q(a)): a\in A; aI \subseteq I_\#\}.
\end{equation*}

\smnoind
(c) Obviously, $xJ_\dc = (0)$ if and only if $xJJ_\dc = (0)$.
Thus, this part follows from part (b) and Theorem \ref{thm:decomp}(d).

\smnoind
(d) Again, as $xJ_\# = (0)$ if and only if $xJJ_\# = (0)$ for $\# = \dc, \tw, \SF$.
This part follows from part (b) as well as Theorem \ref{thm:decomp}(d).
\end{prf}

\medskip

We end this section with the following direct consequence of Theorems \ref{thm:inf-part} and \ref{thm:decomp} (notice that
$I^\bot \cap J^\bot = (I+J)^\bot$).

\medskip

\begin{cor}\label{cor:decomp}
Let $A_{\dc,1} := A_\dc\cap A_1$, $A_{\dc,\infty} := A_\dc\cap A_\infty$,
$A_{\tw,1} := A_\tw\cap A_1$ and $A_{\tw,\infty} := A_\tw\cap A_\infty$.
Then $A_{\dc,1}$, $A_{\dc,\infty}$, $A_{\tw,1}$ and $A_{\tw,\infty}$ are the largest discrete finite ideal, the largest discrete anti-finite ideal, the largest type $\tw$ finite ideal and the largest type $\tw$ anti-finite ideal respectively.
These are annihilator ideals.
They are mutually disjoint and are all disjoint from $A_\ttr$.
Moreover, $A_{\dc,1} + A_{\dc,\infty} + A_{\tw,1} + A_{\tw,\infty} + A_\ttr$ is an essential ideal of $A$, or equivalently,
$$A_{\dc,1} \oplus A_{\dc,\infty} \oplus A_{\tw,1} \oplus A_{\tw,\infty} \oplus A_\ttr \subseteq A \subseteq M(A_{\dc,1}) \oplus M(A_{\dc,\infty}) \oplus M(A_{\tw,1}) \oplus M(A_{\tw,\infty}) \oplus M(A_\ttr).$$
\end{cor}

\medskip

Observe that the $C^*$-algebras $M(A_{\dc,1})$, $M(A_{\dc,\infty})$, $M(A_{\tw,1})$, $M(A_{\tw,\infty})$ and $M(A_\ttr)$ are also finite discrete, anti-finite discrete, finite type $\tw$, anti-finite type $\tw$ and type $\ttr$, respectively (by Proposition \ref{prop:type-hered}(b) and the argument of Theorem \ref{thm:inf-part}).
Furthermore, one may obtain results similar to those in Corollary \ref{cor:quot} for $A_{\dc,1}$, $A_{\dc,\infty}$, $A_{\tw,1}$ and $A_{\tw,\infty}$, but we leave them to the readers.

\bigskip

\section{Prime $C^*$-algebras and $C^*$-algebras with extremely disconnected spectra}

In this section, we will consider $C^*$-algebras for which we have a better decomposition in Corollary \ref{cor:decomp}.
These are the algebras on which every annihilator ideal is complementary.
We will give some equivalent forms of these algebras.
Before that, let us start with the following result.

\medskip

\begin{lem}\label{lem:ann-id-ess-ext}
If $A$ is an essential extension of a $C^*$-algebra $B$, the map $\Gamma:J\mapsto J\cap B$ is a bijection from the set of annihilator ideals of $A$ onto that of $B$.
\end{lem}
\begin{prf}
Let $I\subseteq A$ be a closed ideal.
We first show that
\begin{equation}\label{eqt:cap-bot}
I^\bot = (I\cap B)^\bot.
\end{equation}
Indeed, one clearly has
$I^\bot \subseteq (I\cap B)^\bot$.
Suppose on the contrary that there is $x\in (I\cap B)^\bot \setminus I^\bot$.
Then $xI\neq (0)$ and hence $\overline{IxI}$ is a non-zero ideal of $A$.
As $B$ is an essential ideal of $A$, one can find $b\in \overline{IxI}\cap B\setminus \{0\}$.
Thus, $b$ can be approximated in norm by elements of the form $\sum_{k=1}^n y_kxz_k$ where $y_1,...,y_n, z_1,...,z_n\in I$, and $bb^*$ can be approximated by elements of the form $\sum_{k=1}^n y_kxz_kb^*$.
Since $z_kb^*\in I\cap B$, we obtain the contradiction that $bb^* = 0$.

Now, if $J = I^\bot$, by Relation \eqref{eqt:cap-bot}, we know that $J\cap B = (I\cap B)^\bot \cap B$ and hence $\Gamma(J)$ is an annihilator ideal of $B$.
Furthermore, as
\begin{equation}\label{eqt:J-cap-A-cl}
\Gamma(J)^{\bot\bot} = (J\cap B)^{\bot \bot} = J^{\bot\bot} = J
\end{equation}
(by Relation \eqref{eqt:cap-bot}), we see that $\Gamma$ is injective.

Finally, let $J_0$ be an annihilator ideal of $B$ and  $I_0\subseteq B$ be an ideal satisfying $J_0= I_0^\bot \cap B$.
If $J:= I_0^\bot$, then obviously, $\Gamma(J) = J_0$.
\end{prf}

\medskip

Recall that a topological space (not necessarily Hausdorff) is \emph{extremely disconnected} if the closure of every open subset is again open.
It happens that the $C^*$-algebras described in the beginning of this section are precisely those with extremely disconnected primitive spectra (when equipped with the hull-kernel topology).
We will   give some other equivalent forms of these algebras (although not all of them are needed in this paper).

\medskip

\begin{prop}\label{prop:cent-AW-st}
The following statements are equivalent.
\begin{enumerate}
\item The primitive spectrum $\pr(A)$ is extremely disconnected.
\item $A = I^\bot + I^{\bot\bot}$, for every ideal $I\subseteq A$.
\item $\overline{p}^1\in \op(A)$ for every $p\in \op(A)\cap Z(A^{**})$.
\item $\Lambda(q):= qA$ is a surjection (and hence a bijection) from  $\proj(ZM(A))$ onto the set of non-zero annihilator ideals of $A$.
\item $\pr(M(A))$ is extremely disconnected.
\item $ZM(A)$ is an $AW^*$-algebra and $\Delta: J\mapsto J\cap ZM(A)$ is an injection from the set of annihilator ideals of $M(A)$ 
to the set of annihilator ideals  of $ZM(A)$.
\item $A$ is boundedly centrally closed (in the sense of \cite[p.165]{AM94}).
\end{enumerate}
\end{prop}
\begin{prf}
(a) $1)\Leftrightarrow 2)$.
This follows from the following two facts concerning a closed ideal $I\subseteq A$:  the closure of $\pr (A) \setminus \hull (I)$ equals $\hull (I^\bot)$,
% for every non-zero closed ideal, there is a primitive ideal not containing it;
% every primitive ideal is prime
and the subset $\hull (I^\bot)$ is open if and only if $A = I^\bot + I^{\bot\bot}$.
% by the first fact, ${\rm hull} (I^\bot)$ is open if and only if ${\rm hull}(I^\bot)\cap {\rm hull}(I^{\bot\bot}) = \emptyset$

\smallskip\noindent
$2)\Rightarrow 3)$.
As $\her_A(p)^\bot = \her_A(1-\overline{p}^1)$ (see Remark \ref{rem:essetial}(a)), Statement 2 implies that $\her_A(1-\overline{p}^1) + \her_A(1-\overline{p}^1)^\bot = A$, and hence $1-\overline{p}^1\in M(A)$.

\smallskip\noindent
$3)\Rightarrow 4)$.
Suppose that $I\subseteq A$ is a closed ideal and $J := I^\bot$.
If $p, q\in \op(A)\cap Z(A^{**})$ with $I = \her_A(p)$ and $J = \her_A(q)$, Remark \ref{rem:essetial}(a) tells us that $q = 1-\overline{p}^1$.
As $\overline{p}^1$ is both open and closed, it belongs to $M(A)$ and so is $q$.
Thus, $J = qA$ as required.

\smallskip\noindent
$4)\Rightarrow 2)$.
It is clear that if $\Lambda$ is surjective, then $A = I^\bot + I^{\bot\bot}$.

\smallskip\noindent
$4)\Leftrightarrow 5)$.
By applying the equivalence of Statements 1 and 4 (as established above) to $M(A)$, it suffices to show that Statement 4 (for $A$) is equivalent to
the following statement.
\begin{enumerate}[(1')]
\setcounter{enumi}{3}
\item For every non-zero annihilator ideal $J\subseteq M(A)$, there exists $q\in \proj(ZM(A))$ with $J = qM(A)$.
\end{enumerate}

In fact, suppose that $J$ is a non-zero annihilator ideal of $M(A)$.
By Lemma \ref{lem:ann-id-ess-ext}, $J\cap A$ is an annihilator ideal of $A$.
Hence, Statement 4 (for $A$) gives $q\in \proj(ZM(A))$ with $J\cap A = qA$.
Observe that if $x\in M(A)$ satisfying $xqA = (0)$, then $x =  (1-q)x$.
Thus, Relation \eqref{eqt:J-cap-A-cl} tells us that $J = (qA)^{\bot\bot} = \big((1-q)M(A)\big)^\bot = qM(A)$ (note that these $\bot$ are taken in $M(A)$).

Conversely, if $I$ is a non-zero annihilator ideal of $A$, then Lemma \ref{lem:ann-id-ess-ext} and Statement 4' produce $q\in \proj(ZM(A))$ satisfying $I = qM(A) \cap A = qA$, as is required.

\smallskip\noindent
$1)\Rightarrow 6)$.
Suppose that $\CI$  is a closed ideal of $ZM(A)\cong C_b(\pr(A))$ (the Dauns-Hofmann theorem).
Then $U_\CI:= \{J\in \pr(A): f(J)\neq 0, \text{ for some }f\in \CI\}$ is open and its closure $\overline{U_\CI}$ is an open subset of $\pr(A)$.
One may regard $\CI_0:= C_b(\overline{U_\CI})$ and $\CJ:=C_b(\pr(A) \setminus \overline{U_\CI})$ as disjoint ideals of $C_b(\pr(A))$ with their sum equals $C_b(\pr(A))$.
Thus, $\CJ = \CI_0^\bot$ and $\CI_0 = \CJ^\bot$ (here $\bot$ are taken in $ZM(A)$).
Since $\CI\subseteq \CI_0$ and $\CI^\bot \subseteq \CJ$, we see that $\CI_0 = \CI^{\bot\bot}$ and $\CJ = \CI^\bot$.
Thus, $ZM(A) = \CI^\bot + \CI^{\bot\bot}$ and the spectrum of $ZM(A)$ is extremely disconnected (by the equivalence of Statements 1 and 2, as established above, for $ZM(A)$).

Let $J$ be a non-zero annihilator ideal of $M(A)$.
As $\pr(M(A))$ is extremely disconnected (by the equivalence of Statements 1 and 5), we know that there is $q\in \proj(ZM(A))$ with $J = qM(A)$ (by Statement 4') and $J\cap ZM(A) = qZM(A)$, which is an annihilator ideal of $ZM(A)$.
The injectivity of $\Delta$ follows from the fact that if $p,q\in \proj(ZM(A))\cup \{0\}$ with $pZM(A) = qZM(A)$, then $p=q$.

\smallskip\noindent
$6)\Rightarrow 1)$.
By the equivalences established above, it suffices to verify Statement 4'.
Suppose that $J$ is a non-zero annihilator ideal of $M(A)$.
By the assumption, $J\cap ZM(A)$ is an annihilator ideal and there is $q\in \proj(ZM(A))\cup \{0\}$ with $J \cap ZM(A) = qZM(A)$ (because $ZM(A)$ is a commutative $AW^*$-algebra).
Since $qM(A) \cap ZM(A) = J\cap ZM(A)$, the injectivity of $\Delta$ tells us that $J = qM(A)$.

\smallskip\noindent
$1)\Leftrightarrow 7)$.
This is precisely \cite[Proposition 2.9]{AM94}.
\end{prf}

\medskip

If $\pr(A)$ is extremely disconnected, the map $\Delta$ in Statement 6 is actually a bijection.
On the other hand, in the case when $A$ unital, the implication $1)\Rightarrow 6)$ also follows implicitly from \cite[Theorem 2.1]{Some96} and \cite[Proposition 3.2]{AS90}, but it will be easier to do this directly rather than to recall the definition of ``quasi-standard'' $C^*$-algebras and their properties.

\medskip

As the topology on the spectrum of a $C^*$-algebra $A$ is the quotient topology induced from the hull-kernel topology on $\pr(A)$, we know that $A$ has an extremely disconnected primitive spectrum if and only if $A$ has an extremely disconnected spectrum.
By Proposition \ref{prop:cent-AW-st}, all $AW^*$-algebras as well as all prime $C^*$-algebras have extremely disconnected spectra.
Proposition \ref{prop:cent-AW-st} also tells us that if $A$ has an extremely disconnected spectrum, then $A$ is prime if and only if $\dim ZM(A) = 1$.
Moreover, by \cite[Example 2.10]{AM94}, the $C^*$-algebra $M_{\loc}(A)$ of local multipliers of $A$, is boundedly centrally closed (and hence has an extremely disconnected spectrum).

\medskip

In the following, we denote by $\beta \KI$ the Stone-Cech compactification of a (discrete) set $\KI$.

\medskip

\begin{prop}\label{prop:quasi-Ston}
Let $A$ be a $C^*$-algebra with an extremely disconnected primitive spectrum.

\smnoind
(a) $A = A_{\dc,1} \oplus A_{\dc,\infty} \oplus A_{\tw,1} \oplus A_{\tw,\infty} \oplus A_\ttr$.

\smnoind
(b) Suppose that there is a set $\KI$ with $ZM(A)\cong \ell^\infty(\KI)$.
Then $\KI$ is the disjoint union of subsets $\KI_{\dc,1}$, $\KI_{\dc,\infty}$, $\KI_{\tw,1}$, $\KI_{\tw,\infty}$ and $\KI_\ttr$ such that for $\# = \{\dc,1\}, \{\dc,\infty\}, \{\tw,1\}, \{\tw,\infty\}, \ttr$, there is a  continuous field $\{A_\omega\}_{\omega\in \beta\KI_\#}$ of $C^*$-algebras on $\beta\KI_\#$ with $A_\#$ being the algebra of continuous sections of $\{A_\omega\}_{\omega\in \beta\KI_\#}$.
Moreover, $A_i$ is prime for each $i\in \KI_\#$, the canonical map $\Theta: A_\#\to\prod_{i\in \KI_\#}^{\ell^\infty} A_i$ is injective and $\overline{\sum_{i\in \KI_\#} A_i}$ can be regarded as an essential ideal of $A_\#$.
\end{prop}
\begin{prf}
(a) By Proposition \ref{prop:cent-AW-st}, as well as Theorems \ref{thm:inf-part} and \ref{thm:decomp}, the central open projections corresponding to the ideals $A_{\dc,1}$, $A_{\dc,\infty}$, $A_{\tw,1}$, $A_{\tw,\infty}$ and $A_\ttr$ belong to $M(A)$ with their sum being $1$ (since this sum is dense in $1$, because of Corollary \ref{cor:decomp}).
This gives the conclusion.

\smnoind
(b) Let $p_\#$ be the projection in $ZM(A)$ with $A_\# = p_\# A$.
Then $p_\#\in \ell^\infty(\KI)$ is the indicator function for a subset $\KI_\#$, and $\KI$ is the disjoint union of  $\KI_{\dc,1}$, $\KI_{\dc,\infty}$, $\KI_{\tw,1}$, $\KI_{\tw,\infty}$ and $\KI_\ttr$.
Now, one may regard $A_\#$ as a Banach $\ell^\infty(\KI_\#)$-normed module, and hence  consists of the continuous sections of a continuous field $\{A_\omega\}_{\omega\in \beta\KI_\#}$ of $C^*$-algebras (see e.g.\ \cite{DG83}).

For every $i\in \KI_\#$, we consider $p_i$ to be the projection in $ZM(A)$ corresponding to the point mass at $i$.
Then we have $A_i= p_i A$ and the open projection corresponding to $A_i$ is again $p_i$.
Suppose that $J$ is a non-zero closed ideal of $A_i$ and $p\in \op(A)$ with $J = \her_A(p)$.
As $p\leq p_i$ and $p_i\in M(A)$, we know that $\overline{p}^1 \leq p_i$.
Since $\overline{p}^1\in M(A)$ (by Proposition \ref{prop:cent-AW-st}) and $p_i$ is a minimal projection in $ZM(A)$, we know that $\overline{p}^1 = p_i$ and $J$ is an essential ideal of $A_i$.
This shows that $A_i$ is prime.

Clearly, $A_i$ is an ideal of $A_\#$ and $I_\#:=\overline{\sum_{i\in \KI_\#} A_i}$ is a closed ideal of $A_\#$.
As $A_\#$ also has an extremely disconnected spectrum, the open projection $q$ with $I_\#^\bot= \her_A(q)$ is the indicator function in $\ell^\infty(\KI_\#)$ for a subset $S\subseteq \KI_\#$.
If $S$ is non-empty and $i\in S$, we will have the contradiction that $A_i\subseteq I_\#\cap I_\#^\bot$.
Thus, $I_\#$ is an essential ideal of $A_\#$.

Finally, suppose that $a\in \ker \Theta$.
If we consider $a$ as a continuous section on $\{A_\omega\}_{\omega\in \beta \KI_\#}$, then its values at every $i\in \KI_\#$ is zero and hence $a$ is the zero section (as $\KI_\#$ is dense in $\beta\KI_\#$).
\end{prf}

\medskip

Part (a) of the above tells us that the decomposition in Corollary \ref{cor:decomp} extends the corresponding one for $AW^*$-algebras.
Moreover, part (b) applies to all $AW^*$-algebras whose centers are of the form $\ell^\infty(\KI)$.


\medskip

Because of Proposition \ref{prop:quasi-Ston}(b) as well as the fact that prime $C^*$-algebras are those with no non-trivial annihilator ideal, one may regard prime $C^*$-algebras as ``$C^*$-factors''.
We end this paper by considering some other situations when $C^*$-algebras can be ``decomposed'' as continuous fields of ``$C^*$-factors'', each of which is of a particular type.

\medskip

\begin{prop}\label{prop:ess-simp-decomp}
(a) If $A$ is a prime $C^*$-algebra, then $A$ is of one of the five types: finite discrete, anti-finite discrete, finite type $\tw$, anti-finite type $\tw$ or type $\ttr$.

\smnoind
(b) If $A$ is discrete (respectively, anti-finite or tracially infinite, in the sense of Definition \ref{prop:pure-inf>III}), there is an open dense subset $\Xi_\dc$ (respectively, $\Xi_\infty$ or $\Xi_\tsinf$) of $\pr(A)$ such that $A/P$ is discrete (respectively, anti-finite or tracially infinite) for every $P$ in $\Xi_\dc$ (respectively, $\Xi_\infty$ or $\Xi_\tsinf$).
\end{prop}
\begin{prf}
(a) As said in Corollary \ref{cor:decomp}, $A_{\dc,1}$, $A_{\dc,\infty}$, $A_{\tw,1}$, $A_{\tw,\infty}$ and $A_\ttr$ are annihilator ideals of $A$ and hence $A$ coincides with exactly one of them.

\smnoind
(b) Suppose that $A$ is discrete.
By \cite[Theorem 2.3(v)]{PZ00}, the largest type $\tI$ ideal $A_{\tI}$ is essential.
Thus,
$$\Xi_\dc:=\pr(A)\setminus \hull(A_{\tI})$$
is an open dense subset of $\pr(A)$.
For every $P\in \Xi_\dc$, the image of $A_{\tI}$ in $A/P$
is a non-zero closed ideal of type $\tI$.
Thus, by part (a), $A/P$ is discrete.

Secondly, we consider the case when $A$ is anti-finite.
By Proposition \ref{prop:equiv-anti-fin}(a), there is an essential closed ideal $J_0$ of $A$ with $T_s(J_0) = \emptyset$.
Set
$$\Xi_\infty:= \pr(A)\setminus {\hull}(J_0).$$
Then $\Xi_\infty$ is an open dense subset of $\pr(A)$.
If $P\in \Xi_\infty$ and $q:A\to A/P$ is the quotient map, then $q(J_0)\neq (0)$ and does not have a tracial state.
Thus, $q(J_0)$ is anti-finite and so is the prime $C^*$-algebra $A/P$ (by part (a)).

In the case when $A$ is tracial infinite, there is an essential closed ideal $J_1\subseteq A$ with $\T(J_1) = \{0\}$, and we set
$$\Xi_\tsinf:= \pr(A)\setminus {\hull}(J_1).$$
The argument for this case is similar to the case of anti-finiteness, except that one also needs the argument of Proposition \ref{prop:pure-inf>III}(a) for showing that if $\T(A) = \{0\}$, then $\T(A/I) = \{0\}$ for every closed ideal $I\subseteq A$.
\end{prf}

\medskip

\begin{eg}
Let $A=\CL(\ell^2)$ and $P= \CK(\ell^2)$.
Then $A$ is discrete and $A/P$ is of type $\ttr$.
Therefore, one cannot expect $\Xi_\dc = \pr(A)$ in Proposition \ref{prop:ess-simp-decomp}(b).
\end{eg}

\medskip

By the Dauns-Hofmann theorem, one has a continuous map $\Phi: \pr(A) \to \pr(ZM(A))$ with dense range. %changed after submission: not open map
In fact, if we identify elements in $\pr(ZM(A))$ with one-dimensional unital $^*$-representations of $ZM(A)$, then $\Phi(P)(x) = \|x+\ti P\|_{M(A)/\ti P}$ for every $x\in ZM(A)_+$, where $$\ti P := \{m\in M(A): mA, Am \subseteq P\}$$
is the unique element in $\pr(M(A))$ with $\ti P\cap A = P$.
Thus, $\Phi(P) = \omega$ if and only if $\ker(\omega)\subseteq \ti P$.

\medskip

Let $I$ be a closed ideal of $A$.
It is well-known that $I$ is the algebra of (bounded) continuous sections of a continuous fields $\Upsilon^I$ of $C^*$-algebras over $\pr(ZM(A))$ with the fiber $\Upsilon^I_\omega$ over $\omega\in \pr(ZM(A))$ being $I/\overline{I\ker(\omega)}$ (see e.g.\ \cite{DG83}).

\medskip

The following lemma gives another description for the fiber and it also gives a ``visual description'' of the image of an open subset of $\pr(A)$ under $\Phi$.
This lemma could be known, but since we do not find it in the literature, we give it argument here for completeness.

\medskip

\begin{lem}\label{lem:image-open-dense}
Let $I\subseteq A$ be a non-zero closed ideal.
For every $\omega\in \pr(ZM(A))$, one has $\overline{I\ker(\omega)} = \bigcap\{P\in \pr(A)\setminus \hull(I): \Phi(P) = \omega\}$.
Consequently,
$$\Phi\big(\pr(A)\setminus \hull(I)\big) = \Omega_I:= \big\{\omega\in \pr(ZM(A)): \Upsilon^I_\omega \neq (0)\big\}.$$
% deleted after submission: and $I$ is the algebra of $C_0$-sections of $\Upsilon^I|_{\Omega_I}$.
\end{lem}
\begin{prf}
The second statement clearly follows from the first one.
Thus, we will only establish the first statement.
Since $\overline{I\ker(\omega)}$ is an ideal of $I$, one has $$\overline{I\ker(\omega)} = \bigcap \{P\in \pr(A)\setminus \hull(I): \overline{I\ker(\omega)}\subseteq P\}.$$
Now, pick any $\omega\in \Omega$ and $P\in \pr(A)\setminus \hull(I)$.
Obviously, if $\omega = \Phi(P)$, then $\ker(\omega)\subseteq \ti P$ and we have  $I\ker(\omega)\subseteq P$.
Conversely, suppose that $\overline{I\ker(\omega)}\subseteq P$.
Let $Q:= P\cap I\in \pr(I)$ and
$$\ti Q:= \{m\in M(I): mI,Im\subseteq Q\}.$$
Consider $\Psi: M(A)\to M(I)$ to be the canonical $^*$-homomorphism.
Set $J:= \overline{A\Psi^{-1}\big(\ti Q\big)}$.
As $\overline{I\ker(\omega)}\subseteq P$, one has $\Psi(\ker(\omega))\subseteq \ti Q$ and $A\ker(\omega)\subseteq J$.
Moreover, one has $\Psi(IJ)\subseteq Q$ and hence $IJ \subseteq Q\subseteq P$ (since $\Psi$ is an injection on $I$).
This implies that $J\subseteq P$ (as $I\nsubseteq P$ by the assumption).
Thus, $A\ker(\omega) \subseteq P$, which implies that $\ker(\omega)\subseteq \ti P$ and so, $\omega = \Phi(P)$.
\end{prf}

\medskip

As in \cite[p.351]{AS90} and \cite[p.125]{Some96}, for each $\omega\in \pr(ZM(A))$, we call $G(\omega):=\bigcap \Phi^{-1}(\omega)$ a \emph{Glimm ideal} of $A$ (here we use the usual convention that $\bigcap \emptyset = A$).

\medskip

\begin{thm}\label{thm:Glimm>prime}
Suppose that the map $\Phi$ is open and every non-zero Glimm ideal of a $C^*$-algebra $A$ is a prime ideal.

\smnoind
(a) $A$ is the algebra of $C_0$-sections of a continuous field of $C^*$-algebras over $\Omega_A:=\Phi(\pr(A))$ with each fiber being non-zero and of one of the five types.

\smnoind
(b) If $A$ is discrete (respectively, anti-finite or tracially infinite), there is an open dense subset $\Omega_\dc$ (respectively, $\Omega_\infty$ or $\Omega_\tsinf$) of $\Omega_A$ such that the fiber over whose elements are discrete (respectively, anti-finite or tracially infinite).
\end{thm}
\begin{prf}
(a) By Lemma \ref{lem:image-open-dense}, one has $G(\omega) = \overline{A\ker(\omega)}$ for every $\omega\in \Omega_A$.
Hence, this part follows directly from Lemma \ref{lem:image-open-dense}, Proposition \ref{prop:ess-simp-decomp}(a) and the assumption on $G(\omega)$.

\smnoind
(b) Suppose that $A$ is discrete, $A_{\tI}$ is the largest type $\tI$ closed ideal of $A$ and $\Xi_\dc:= \pr(A)\setminus {\hull}(A_{\tI})$.
By the proof of Proposition \ref{prop:ess-simp-decomp}(b), $\Xi_\dc$ is an open dense subset of $\pr(A)$.
Set $\Omega_\dc:=\Phi(\Xi_\dc)$. 
Then $\Omega_\dc$ is an open dense subset of $\Omega_A$ satisfying
\begin{equation*}\label{eqt:Omega-d}
\Omega_\dc\subseteq \{\omega\in \Omega_A: G(\omega)\subseteq P, \text{ for some } P\in \Xi_\dc\}
\end{equation*}
(since $G(\omega)\subseteq P$ whenever $P\in \Phi^{-1}(\omega)$).

Pick any $\omega\in \Omega_d$.
If  $A_{\tI}$ is contained in $G(\omega)$, then we know that $A_{\tI}\subseteq P$ for some $P\in \pr(A)\setminus {\hull}(A_{\tI})$, which is absurd.
Consequently, for every $\omega\in \Omega_d$, the image of $A_{\tI}$ in the prime $C^*$-algebra $A/G(\omega)$ is a non-zero type $\tI$ closed ideal, which implies that $A/G(\omega)$ is discrete (because of Proposition \ref{prop:ess-simp-decomp}(a)).

The argument for the anti-finite case and the tracially infinite case are similar.
\end{prf}

\medskip

If $\pr(A)$ is Hausdorff, then $\Phi$ is injective and the hypothesis of Theorem \ref{thm:Glimm>prime} is satisfied.
Moreover, if $A$ is a ``standard'' $C^*$-algebras as in \cite[p.127]{Some96}, then the two hypothesis in Theorem \ref{thm:Glimm>prime} are satisfied (see also \cite[Theorem 3.4]{AS90}). %changed after submission
Thus, Theorem \ref{thm:Glimm>prime} gives the following corollary, thanks to \cite[p.127 and Theorem 2.4]{Some96}.

\medskip

\begin{cor}\label{cor:ess-simp}
If either $\pr(A)$ is Hausdorff, or $A$ is an $AW^*$-algebra, or $A$ is the local multiplier algebra of another $C^*$-algebra $B$, then $A$ is the algebra of continuous $C_0$-sections of a continuous fields of prime $C^*$-algebras over a locally compact Hausdorff space $\Omega_A$ with each fiber being of one of the five types.
If, in addition, $A$ is discrete (respectively, anti-finite or tracially infinite), there is an open dense subset of $\Omega_A$ on which the fiber are 
nonzero and discrete (respectively, anti-finite or tracially infinite).
\end{cor}

\medskip


\bibliographystyle{plain}
\begin{thebibliography}{99}
\bibitem{Akemann69}
C.\ A.\ Akemann, The general Stone-Weierstrass problem, J.\ Funct.\ Anal., \textbf{4} (1969), 277-294.

\bibitem{Akemann70}
C.\ A.\ Akemann, Left ideal structure of $C^*$-algebras, J.\ Funct.\ Anal., \textbf{6} (1970), 305-317.

\bibitem{AM94}
P.\ Ara and M.\ Mathieu, On the central Haagerup tensor product, Proc.\ Edin.\ Math.\ Soc., \textbf{37} (1994), 161-174.

\bibitem{AS90}
R.\ J.\ Archbold and D.\ W.\ B.\ Somerset, Quasi-standard C*-algebras, Math.\ Proc.\ Comb.\ Phil.\ Soc., \textbf{107} (1990), 349-360.

\bibitem{Blac}
B.\ Blackadar, Operator algebras - theory of $C^*$-algebras and $W^*$-algebras, in \emph{Operator Algebras and Non-commutative Geometry III},
Encyc.\ Math.\ Sci., \textbf{122}, Springer-Verlag, Berlin (2006).

\bibitem{BP91}
L.\ G.\ Brown and G.\ K.\ Pedersen,
$C^*$-algebras of real rank zero, J.\ Funct.\ Anal., \textbf{99} (1991), 131-149.

\bibitem{CP79}
J.\ Cuntz and G.\ K.\ Pedersen,
Equivalence and traces on $C^*$-algebras, J.\ Funct.\ Anal., \textbf{33} (1979), 135-164.

\bibitem{Dix77}
J. Dixmier, \emph{$C^*$-Algebras}, North-Holland Publishing (1977).

\bibitem{DG83}
M.\ J.\ Dupr\'{e} and R.\ M.\ Gillette, \emph{Banach bundles, Banach
modules and automorphisms of $C\sp{*} $-algebras}, Research Notes in
Mathematics 92, Pitman (1983).

\bibitem{Fell61}
J.\ M.\ G.\ Fell, The structure of algebras of operator fields, Acta Math.\, \textbf{106} (1961), 233-280.

\bibitem{KRII}
R.\ V.\ Kadison and J.\ R.\ Ringrose, \emph{Fundamentals of the theory of operator algebras Vol. II: Advanced theory}, Pure and Applied Mathematics \textbf{100}, Academic Press (1986).

\bibitem{KR00}
E.\ Kirchberg and M.\ R{\o}rdam, Non-simple purely
infinite $C^*$-algebras, Amer.\ J.\ Math., \textbf{122} (2000), 637--666.

\bibitem{Lan}
E.\ C.\ Lance, \emph{Hilbert $C^*$-modules - A toolkit for operator algebraists}, Lond.\ Math.\ Soc.\ Lect.\ Note Ser.\ \textbf{210},  Camb.\ Univ.\ Press (1995).

\bibitem{LinHX90}
H.\ Lin, Equivalent open projections and corresponding hereditary
$C^*$-subalgebras, J.\ Lond.\ Math.\ Soc., \textbf{41}
(1990), 295-301.

%\bibitem{Murphy}
%G.J. Murphy, \emph{$C^*$-algebras and operator theory}, Academic Press (1990).

\bibitem{Murray90}
F.\ J.\ Murray,
The rings of operators papers,
in \emph{The legacy of John von Neumann} (Hempstead, NY, 1988), 57-60, Proc.\ Sympos.\ Pure Math.\ \textbf{50}, Amer.\ Math.\ Soc., Providence, R.I.\ (1990).

\bibitem{Murray-vonNeumann36}
F.\ J.\ Murray and J.\ Von Neumann,
On rings of operators,
Ann.\ of Math.\ (2), \textbf{37} (1936), 116-229.

\bibitem{Ng-str-amen-rep}
C.\ K.\ Ng, Strictly amenable representations of reduced group $C^*$-algebras, preprint.

\bibitem{Ng-ext-lscsf-tr}
C.\ K.\ Ng, On extension of lower semi-continuous semi-finite traces, in preparation.

\bibitem{NW-Mv-class}
C.\ K.\ Ng and N.\ C.\ Wong, A Murray-von Neumann type classification of $C^*$-algebra, preprint (arXiv:1112.1455).

%\bibitem{NW-ext-v-alg-class}
%C. K. Ng and N. C. Wong, An extension of $W^*$-algebra classification to $C^*$-algebras, preprint.

%\bibitem{ORT}
%E.\ Ortega, M.\ R{\o}rdam and H.\ Thiel,
%The Cuntz semigroup and comparison of open projections, J.\ Funct.\ Anal., \textbf{260} (2011), 3474--3493.

\bibitem{Ped79}
G.\ K.\ Pedersen,
\emph{$C^*$-algebras and their automorphism groups},
Academic Press (1979).

\bibitem{PZ00}
C.\ Peligrad and L.\ Zsid\'{o}, Open projections of $C^*$-algebras:
Comparison and Regularity, \emph{Operator Theoretical Methods, 17th
Int.\ Conf.\ on Operator Theory, Timisoara (Romania), June 23-26,
1998}, Theta Found.\ Bucharest (2000), 285-300

\bibitem{Some96}
D.\ W.\ B.\ Somerset, The local multiplier algebra of a C*-algebra, Quart.\ J.\ Math.\ Oxford, Ser.\ 2, \textbf{47} (1996), 123-132.

\bibitem{Zhang89}
S.\ Zhang, Stable isomorphism of
hereditary $C^\ast$-subalgebras and stable equivalence of open projections,
Proc.\ Amer.\ Math.\  Soc., \textbf{105} (1989), 677-682.
\end{thebibliography}

\end{document}

1310.5464.extracted_bib
\begin{thebibliography}{99}
\bibitem{Akemann69}
C.\ A.\ Akemann, The general Stone-Weierstrass problem, J.\ Funct.\ Anal., \textbf{4} (1969), 277-294.

\bibitem{Akemann70}
C.\ A.\ Akemann, Left ideal structure of $C^*$-algebras, J.\ Funct.\ Anal., \textbf{6} (1970), 305-317.

\bibitem{AM94}
P.\ Ara and M.\ Mathieu, On the central Haagerup tensor product, Proc.\ Edin.\ Math.\ Soc., \textbf{37} (1994), 161-174.

\bibitem{AS90}
R.\ J.\ Archbold and D.\ W.\ B.\ Somerset, Quasi-standard C*-algebras, Math.\ Proc.\ Comb.\ Phil.\ Soc., \textbf{107} (1990), 349-360.

\bibitem{Blac}
B.\ Blackadar, Operator algebras - theory of $C^*$-algebras and $W^*$-algebras, in \emph{Operator Algebras and Non-commutative Geometry III},
Encyc.\ Math.\ Sci., \textbf{122}, Springer-Verlag, Berlin (2006).

\bibitem{BP91}
L.\ G.\ Brown and G.\ K.\ Pedersen,
$C^*$-algebras of real rank zero, J.\ Funct.\ Anal., \textbf{99} (1991), 131-149.

\bibitem{CP79}
J.\ Cuntz and G.\ K.\ Pedersen,
Equivalence and traces on $C^*$-algebras, J.\ Funct.\ Anal., \textbf{33} (1979), 135-164.

\bibitem{Dix77}
J. Dixmier, \emph{$C^*$-Algebras}, North-Holland Publishing (1977).

\bibitem{DG83}
M.\ J.\ Dupr\'{e} and R.\ M.\ Gillette, \emph{Banach bundles, Banach
modules and automorphisms of $C\sp{*} $-algebras}, Research Notes in
Mathematics 92, Pitman (1983).

\bibitem{Fell61}
J.\ M.\ G.\ Fell, The structure of algebras of operator fields, Acta Math.\, \textbf{106} (1961), 233-280.

\bibitem{KRII}
R.\ V.\ Kadison and J.\ R.\ Ringrose, \emph{Fundamentals of the theory of operator algebras Vol. II: Advanced theory}, Pure and Applied Mathematics \textbf{100}, Academic Press (1986).

\bibitem{KR00}
E.\ Kirchberg and M.\ R{\o}rdam, Non-simple purely
infinite $C^*$-algebras, Amer.\ J.\ Math., \textbf{122} (2000), 637--666.

\bibitem{Lan}
E.\ C.\ Lance, \emph{Hilbert $C^*$-modules - A toolkit for operator algebraists}, Lond.\ Math.\ Soc.\ Lect.\ Note Ser.\ \textbf{210},  Camb.\ Univ.\ Press (1995).

\bibitem{LinHX90}
H.\ Lin, Equivalent open projections and corresponding hereditary
$C^*$-subalgebras, J.\ Lond.\ Math.\ Soc., \textbf{41}
(1990), 295-301.

%\bibitem{Murphy}
%G.J. Murphy, \emph{$C^*$-algebras and operator theory}, Academic Press (1990).

\bibitem{Murray90}
F.\ J.\ Murray,
The rings of operators papers,
in \emph{The legacy of John von Neumann} (Hempstead, NY, 1988), 57-60, Proc.\ Sympos.\ Pure Math.\ \textbf{50}, Amer.\ Math.\ Soc., Providence, R.I.\ (1990).

\bibitem{Murray-vonNeumann36}
F.\ J.\ Murray and J.\ Von Neumann,
On rings of operators,
Ann.\ of Math.\ (2), \textbf{37} (1936), 116-229.

\bibitem{Ng-str-amen-rep}
C.\ K.\ Ng, Strictly amenable representations of reduced group $C^*$-algebras, preprint.

\bibitem{Ng-ext-lscsf-tr}
C.\ K.\ Ng, On extension of lower semi-continuous semi-finite traces, in preparation.

\bibitem{NW-Mv-class}
C.\ K.\ Ng and N.\ C.\ Wong, A Murray-von Neumann type classification of $C^*$-algebra, preprint (arXiv:1112.1455).

%\bibitem{NW-ext-v-alg-class}
%C. K. Ng and N. C. Wong, An extension of $W^*$-algebra classification to $C^*$-algebras, preprint.

%\bibitem{ORT}
%E.\ Ortega, M.\ R{\o}rdam and H.\ Thiel,
%The Cuntz semigroup and comparison of open projections, J.\ Funct.\ Anal., \textbf{260} (2011), 3474--3493.

\bibitem{Ped79}
G.\ K.\ Pedersen,
\emph{$C^*$-algebras and their automorphism groups},
Academic Press (1979).

\bibitem{PZ00}
C.\ Peligrad and L.\ Zsid\'{o}, Open projections of $C^*$-algebras:
Comparison and Regularity, \emph{Operator Theoretical Methods, 17th
Int.\ Conf.\ on Operator Theory, Timisoara (Romania), June 23-26,
1998}, Theta Found.\ Bucharest (2000), 285-300

\bibitem{Some96}
D.\ W.\ B.\ Somerset, The local multiplier algebra of a C*-algebra, Quart.\ J.\ Math.\ Oxford, Ser.\ 2, \textbf{47} (1996), 123-132.

\bibitem{Zhang89}
S.\ Zhang, Stable isomorphism of
hereditary $C^\ast$-subalgebras and stable equivalence of open projections,
Proc.\ Amer.\ Math.\  Soc., \textbf{105} (1989), 677-682.
\end{thebibliography}