The ideal envelope of an operator algebra
David P. Blecher and Masayoshi Kaneda
A left ideal of any $C^*$-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in \cite{Bid1} that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a $C^*$-algebra. We study operator algebras and their multiplier algebras from the perspective of `Hamana theory' and using the multiplier algebras introduced by the first author.
math0111122.tex
\documentclass[12pt]{amsart}
\textwidth=6.8in
\oddsidemargin=0in
\evensidemargin=0in
\topmargin=0in
\textheight=8.5in
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}}
\renewcommand{\proof}{\underline{Proof.}\quad}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\newcommand{\Co}{\mbox{$\mathbb{C}$}}
\newcommand{\N}{\mbox{$\mathbb{N}$}}
\newcommand{\F}{\mbox{$\mathbb{F}$}}
\newcommand{\Te}{\mbox{$\mathbb{T}$}}
\newcommand{\Z}{\mbox{$\mathbb{Z}$}}
\newcommand{\K}{\mbox{$\mathbb{K}$}}
\newcommand{\Di}{\mbox{$\mathbb{D}$}}
\newcommand{\Bd}{\mbox{$\mathbb{B}$}}
\newcommand{\A}{\mbox{${\mathcal A}$}}
\newcommand{\B}{\mbox{${\mathcal B}$}}
\newcommand{\C}{\mbox{${\mathcal C}$}}
\newcommand{\D}{\mbox{${\mathcal D}$}}
\newcommand{\E}{\mbox{${\mathcal E}$}}
\newcommand{\G}{\mbox{${\mathcal G}$}}
\newcommand{\He}{\mbox{${\mathcal H}$}}
\newcommand{\Ne}{\mbox{${\mathcal N}$}}
\newcommand{\R}{\mbox{${\mathcal R}$}}
\newcommand{\Sy}{\mbox{${\mathcal S}$}}
\newcommand{\Li}{\mbox{${\mathcal L}$}}
\newcommand{\M}{\mbox{${\mathcal M}$}}
\newcommand{\be}{\mbox{${\mathbb B}$}}
\newcommand{\T}{\mbox{${\mathcal T}$}}
\newcommand{\U}{\mbox{${\mathcal U}$}}
 
\begin{document}
   \title[The ideal envelope of an operator algebra]
%Left ideals and approximate identities in operator algebras]
{The ideal envelope of an operator algebra}
 %s, multipliers, and approximate identities in operator algebras}
\author{David P. Blecher and Masayoshi Kaneda} 

\thanks{*This research was 
supported by a grant from the National Science Foundation}  
%\thanks{This draft is 
%probably going to be continually changing 
%for a while.}
\thanks{Revision of Oct 8, 2001. }  
\address{Department of Mathematics\\University of Houston\\
4800 Calhoun\\Houston, TX 77204-3008, U.S.A.} 
\email{dblecher@math.uh.edu and kaneda@math.uh.edu}\maketitle
 
 
\vspace{10 mm}
 
\begin{abstract}
A left ideal of any $C^*$-algebra is an example of an 
operator algebra with a right contractive approximate 
identity (r.c.a.i.).   Conversely, we show here
and in \cite{Bid1} that 
operator algebras with r.c.a.i. should be studied in
terms of a certain left ideal of a $C^*$-algebra.   
We study operator algebras and 
their multiplier algebras from the perspective of `Hamana
theory' and using the multiplier algebras introduced by 
the first author.  
\end{abstract}
 
 
 
\pagebreak
\newpage
 
 
\section{Introduction and notation}
A left ideal of any $C^*$-algebra is an example of an
operator algebra with a right contractive approximate
identity.
 Conversely, we study
operator algebras with  right contractive approximate identity in
terms of a certain left ideal of a $C^*$-algebra. 
 
A (concrete) operator algebra is a subalgebra of $B(H)$,
for some Hilbert space $H$.   More abstractly,
an operator algebra will be an algebra $A$ with
a norm defined on the space $M_n(A)$ of $n \times n$ 
matrices with entries in $A$, for each $n \in \N$, such that
there exists a {\em completely isometric}\footnote{A map 
$T : X \rightarrow Y$ is 
completely isometric if $[x_{ij}] \mapsto
[T(x_{ij})]$ is isometric on $M_n(X)$ for all $n \in \N$} homomorphism
$A \rightarrow B(H)$ for some Hilbert space $H$.
In this paper all our operator algebras and spaces will be taken
to be complete.  We shall say that an 
operator algebra is {\em unital} if it has a two-sided
contractive identity.   
%Unital operator algebras were 
%characterized abstractly in \cite{BRS}.  
In the present paper
we are concerned with operator algebras
with a one-sided (usually right) contractive approximate
identity. 
% An abstract characterization of these 
%algebras was given recently in \cite{BMDO}, but we shall
%not need this characterization here.  
We shall 
abbreviate `right (resp. left) contractive approximate  identity'
to `r.c.a.i.' (resp. `l.c.a.i.').   
%It is well known even for
%Banach algebras that an algebra with both a left and a right
%c.a.i. has a two-sided c.a.i..            
   
%Much of this paper is concerned with left ideals
%in unital operator algebras which have
%a r.c.a.i..    In a formal sense this coincides with the 
%class of all operator algebras with r.c.a.i., since every
%operator algebra is easily seen to be a left ideal in some
%operator algebra.  By Proposition 6.4 in
%\cite{BEZ} (although we shall not use this fact except for
%motivational purposes), this also coincides with the 
%class of left M-ideals in
%unital operator algebras.   We shall not however use 
%this last fact, except as evidence
%that it suggests 
%that these objects are more important than they seem at first.
%We will make the blanket convention in 
%this paper
%that all ideals, left or otherwise, are closed, i.e. complete.
%unital operator algebra (for example if $A \subset B(H)$ 
%consider $\{ T \in B(H) : T A \subset A \}$).   Thus
%in one sense studying
%left ideals with r.c.a.i. is the same as studying
%all operator algebras with r.c.a.i..   By Proposition 6.4 in
%\cite{BEZ}, this also coincides with the class of left M-ideals in
%unital operator algebras.

%We are not aware of any general results in the literature on 
%systematic attempt to study 
%operator algebras with r.c.a.i.,
%and one of the main goals of this paper 
%and its predecessor  \cite{Bid1} is to take on such a study.
%(besides the main example of left ideals in a 
%$C^*$-algebra), they do arise very naturally
%(e.g. $Ae$ for a projection $e$ in a
%unital operator algebra $A$, for example 
%\ref{Exma}; or Example ADDREF in \cite{Bid1}, or
%Theorem 4.11 in \cite{BMP}).   
%The left M-ideal result 
%mentioned in the last paragraph is additional evidence
%that these objects are more important than they seem at first.
%We also felt that it was 
%worthwhile to dispel the common feeling that only 
%operator algebras with 2-sided approximate identities have 
%any satisfactory theory.   Indeed we showed in  \cite{Bid1} amongst other
%things that they 
%have an abstract characterization, Banach-Stone type theorems,
%and they have an
%operator space predual if and only if they are
%dual operator algebras' in the usual strong sense 
%of that term.   Also, this subject becomes a little 
%more interesting with 
In \S 2 of our paper we 
consider a certain `transference principle',
% in mind
%.  This principle is perhaps the most useful
%result of our paper,
which can allow one to deduce many general results about 
 operator algebras  with r.c.a.i., from results about
left ideals in a $C^*$-algebra (see the companion 
paper \cite{Bid1}).  Namely there is an important 
left ideal ${\mathfrak J}_e(A)$ of a $C^*$-algebra $\E(A)$, which 
is associated to any 
such operator algebra.  We call ${\mathfrak J}_e(A)$ the `left ideal 
envelope' of $A$.   This is analoguous to 
the case of operator algebras with 2-sided c.a.i., 
which are largely studied these days
in terms of a certain $C^*$-algebra, namely
the $C^*-$envelope.   

%Because however 
%operator algebras with r.c.a.i. do not constitute 
%an area attracting a  huge amount of interest right now
% we will not go to great lengths to be
%exhaustive here, or to develop any kind of complete
%account of their theory.  Indeed 
%we do not quite get to the bottom of a couple of the topics
%studied here.
%%; and at least right now
%cannot justify spending much more time in the attempt.
    
% \vspace{4 mm}

%We now describe the layout of the paper.  
%In \S 2 we discuss the 
%left ideal envelope, the heart of the 
%`transference principle' mentioned above.
In \S 3  we study the `left multiplier operator algebra' 
%$LM(A)$
  of an  operator algebra $A$ with r.c.a.i. (which will be 
a symmetrical theory to that of $RM(A)$ for an 
operator algebra $A$ with l.c.a.i).   
 This theory does not work out quite 
as nicely as the case when $A$ has a 
two-sided c.a.i., unless $A$ is a
left ideal of a $C^*$-algebra.  We present some 
examples to show that some of the obvious candidate 
descriptions of this multiplier algebra are defective.      
A main motivation for studying multiplier algebras is because of
their intimate connection with the extremely important notion of the 
`unitization' of a nonunital algebra.  A good `unitization' procedure
should reduce many problems about  nonunital algebras to the well
understood unital case.   Although we are not at this point able
to completely analyze the `unitization' procedure for algebras
with one-sided c.a.i., we at least do some of the groundwork 
here.   

%Often it is convenient to state only the `right-handed' version of a
%result, say.  For example Lemma \ref{namo} is a result about
%operator algebras
%with r.c.a.i..   Of course by symmetry
%there will be a matching `left-handed' version, in our example
%it will be about operator algebras with l.c.a.i..  If we want to
%invoke this `left-handed' version, we will refer to the
%`other-handed  version of Lemma \ref{namo}', for example.  Sometimes we will be
%sloppy and simply refer to Lemma \ref{namo}, and leave it to the reader
%to check that the result referred to does indeed have a symmetric
%or `other-handed  version'. 

We end the introduction with some
more notation, and some background results which will
be useful in various places.  
We reserve the letters $H, K$ for Hilbert spaces,
and $J$ for a left ideal of a $C^*$-algebra.
If $A$ is an algebra then we write $\lambda
: A \rightarrow Lin(A)$ for the  canonical `left regular
representation' of $A$ on itself.
%, and $\rho$ for the  `right regular antirepresentation'.  
If $S$ is a 
subalgebra of $A$ then 
the left idealizer of $S$ is the 
subalgebra $\{ x \in A : x S \subset S \}$ of $A$.
Note $S$ is a left ideal in this subalgebra, whence the name.
Similarly for the right idealizer; the (2-sided) 
idealizer is the intersection of the left and right idealizer.

By a `representation' $\pi : A \rightarrow B(H)$ of 
an operator algebra $A$ we shall
mean a completely contractive homomorphism.  If
 $A$ has r.c.a.i. and if we say that 
%we will usually say that 
$\pi$ is {\em nondegenerate}, then at the very least  
we mean that 
$\pi(A) H$ is dense in $H$.  Note that this last 
condition does not imply
in general
 that $\pi(e_\alpha) \zeta \rightarrow \zeta$ for $\zeta \in H$,
where $\{ e_\alpha \}$ is the r.c.a.i.,
as one is used to in the two-sided case.  
%One also cannot appeal to
%Cohen's factorization theorem in its usual form
%(see however \cite{Pal} \S 5.2).    

\begin{lemma} \label{mano}  Let $A$ be an operator algebra with
either a l.c.a.i. or a r.c.a.i..  Suppose that $\pi :
A \rightarrow B(H)$ is a completely contractive
representation,
set $K$ to be the norm closure of $\pi(A) H$, and let
 $\pi' : A \rightarrow B(K)$ be the induced representation.
Then $\pi'$ is a
completely contractive homomorphism, and
it is nondegenerate in the sense that $\pi'(A) K$ is dense
in $K$.  Moreover if $A$ has a r.c.a.i. and
$\pi$  is isometric (resp. completely isometric, 1-1),
then $\pi'$ is also  isometric (resp. completely isometric,
1-1).
 
Consequently, if $A$ is a subalgebra of $B(H)$ with a
r.c.a.i.,  then we may view $A$ as a nondegenerate
subalgebra of $B(K)$, where $K$ is the closure of $A H$.
\end{lemma}
 
\begin{proof}  That $\pi'$ is a completely contractive
homomorphism is clear.  Clearly $\pi'(A) K \subset K$, however
if $a \in A$ and $\zeta \in H$ then $\pi(a) \zeta =
\lim \pi'(a) \pi(e_{\alpha}) \zeta \in
\pi'(A) K$, if $A$ has a r.c.a.i..   Taking linear combinations
of such expressions $\pi(a) \zeta$, and limits,
shows that any element of $K$ is in the closure of $\pi'(A) K$.
So $\pi'$ is nondegenerate in this case. A similar argument
holds if $A$ has a l.c.a.i..
 
To see the statement here about the isometry note that for $a \in A$
we have
$$\Vert a \Vert  = \Vert \pi(a) \Vert = \sup \{ \Vert \pi(a) \zeta \Vert :
\zeta \in Ball(H) \} =
\sup \lim_\alpha
 \{ \Vert \pi'(a) \pi(e_\alpha) \zeta \Vert :
\zeta \in Ball(H) \} \;.$$
However the right hand side is dominated by
$\Vert \pi'(a) \Vert$, so that $
\pi'$ is an isometry.  A similar argument works for a
complete isometry.  The 1-1 assertion is easier.
%$a \eta = \lim a e_{\alpha} \eta$
\end{proof}
 
\vspace{4 mm}
 
If an operator algebra has only a l.c.a.i.
then the
 `isometric' assertions of the last result are not true
in general.  For a counterexample consider $A = R_n$,
the subalgebra of $M_n$ supported on
the first row.   
%Indeed notice that
%if $A$ has a left identity of norm 1 but no right identity,
%and if $\pi : A \rightarrow B(H)$ is a nondegenerate (in
%the sense above)
%isometric representation, then $\pi(e) = Id$,

We will use without comment several very basic facts from
$C^*$-algebra theory (see e.g. \cite{Ped}), such as
the basic definitions
of the left multiplier algebra $LM(A)$,
and the multiplier algebra $M(A)$, of a $C^*$-algebra.
As a general reference for operator spaces the reader might consult
\cite{ERbook,Pis} or \cite{P}.
%, currently the only book on the market on this subject
%(although there are forthcoming books by other authors).
We write $\; \hat{ } \; : X \rightarrow X^{**}$ for the 
canonical map, this is a complete isometry if $X$ is an
operator space, and is a homomorphism if $X$ is an operator 
algebra, giving the second 
dual the Arens product \cite{BonsallDuncan}.

We will often consider the basic examples $C_n$ (resp. $R_n$)
of operator algebras with right (resp. left) identity of norm 
1; namely the $n \times n$ matrices
`supported on' the first column (resp. row).  This is a
left (resp. right) ideal of $M_n$, and has the projection
$E_{11}$ as the 1-sided identity. 

If $X$ and $Y$ are subsets of an operator algebra we usually write 
$X Y$ for the {\em norm closure} of the set of finite 
sums of products $x y$ of a term in $X$ and a term in $Y$.
For example, if $J$ is a left ideal of a $C^*$-algebra $A$,
then with this convention $J^* J$ will be
 a norm closed $C^*$-algebra.  This convention extends to 
three sets, thus $J J^* J = J$ for a left ideal of a $C^*$-algebra
as is well known.  These facts follow easily from the 
well known results that
a (norm closed) left ideal $J$ in a $C^*$-algebra has a
positive right contractive approximate
identity.   Also $J \cap J^* = J^* J \subset J \subset J J^*$,
 so that $J$ is a left ideal of $J J^*$.  
%It is well known (see \cite{Ped},
%although some of 
%this follows also from results in \S 2 below) for a left ideal of a
%$C^*$-algebra that 
%$$J \cap J^* = J^* J \subset J \subset J J^*$$
%so that $J$ is a left ideal of $J J^*$.   
 
For the purposes of this paper
we will define a {\em triple system} to be a norm closed
subspace $X$ of a
$C^*$-algebra such that $X X^* X \subset X$.  By `subspace' we 
will allow for example spaces such as $B(K,H)$, regarded 
as the `1-2-corner' of the $C^*$-algebra 
$B(H \oplus K,H \oplus K)$ in the 
usual way.  It is well known that 
triple systems are `the same thing'
as Hilbert $C^*$-modules, although there is a slight difference 
of emphasis in the two theories; 
the important 
structure on a triple system is the `triple product' 
$x y^* z$.  A `triple subspace' is a 
norm closed vector subspace of a triple system which is closed 
under this triple product.
%operator space ...
% ({\em ternary ring of 
%operators}) is a (norm closed for this paper) subspace $X$
%of $B(K,H)$ such that $X X^* X \subset X$.   
%It is well known
%(copy the proof of Lemma \ref{lidci} below) that in this case 
%$X X^* X = X$.  
If $X$ is a triple system
then $X X^*$ and $X^* X$ are $C^*$-algebras,
which we will call the left and right $C^*$-algebras of $X$ respectively,
and $X$ is a $(X X^*) - (X^* X)$-bimodule.
  A linear map
$T : X \rightarrow Y$ between triple systems is a {\em triple 
morphism} if $T(x y^* z) = T(x) T(y)^* T(z)$ for all 
$x,y,z \in X$.   Triple systems are operator spaces, and 
triple morphisms are completely contractive, and indeed are 
completely isometric if they are 1-1 (see \cite{Ham2},
this is related to results of Harris and Kaup).
A completely
isometric surjection between TRO's is a triple morphism.  This 
last result might date back to around 1986, to Hamana and 
Ruan's PhD thesis independently.
See \cite{Ham2} or \cite{BShi} A.5 for a proof.

Several times in \S 2 we will refer to 
Hamana's triple envelope
$\T(X)$ of an operator space $X$.  This theory 
may be found in \cite{Ham2}, although we review the
 construction of $\T(X)$ briefly in \S 1 below.   
An alternative account of Hamana's theory of 
the triple envelope is given in \cite{BShi}, 
particularly the introduction and Appendix A there.
%and the reader
%is often referred to that source in addition 
%to \cite{Ham2} since the latter may 
%not be readily accessible.
%See also the discussion before \ref{product}.   
The space
$\T(X)$ may be viewed as a triple system, and there is a canonical 
complete isometry $i : X \rightarrow \T(X)$ such that 
there is no nontrivial
triple subspace of $\T(X)$ containing $i(X)$.  We
 write $\E(X)$ and ${\mathcal F}(X)$ for the
left- and right- $C^*$-algebras of $\T(X)$ respectively.     

Next we recall 
the left multiplier algebra
$\M_\ell(X)$ of an operator space $X$.   This is a unital
operator algebra, which may be viewed as a subalgebra of 
$CB(X)$ containing $Id_X$, but with a different (bigger in general)
norm.   Here $CB(X)$ is the space of completely bounded 
linear maps on $X$.
Our first definition of $\M_\ell(X)$ from 
\cite{BShi} was as the linear maps $X \rightarrow X$ corresponding to 
elements $T$ in $LM(\E(X))$ such that $T i(X) \subset i(X)$.
The norm  on $\M_\ell(X)$ is the 
$LM(\E(X))$ norm, and similarly for matrix norms.
There are several other
equivalent definitions of $\M_\ell(X)$ given in
\cite{BShi,BEZ,BPnew} - it is best to view operator space multipliers 
as a `sequence of equivalent definitions'.   In any particular context
one or other of these definitions may be more appropriate.  We 
will make much use the following two simple results from \cite{Bid1}:          
 
\begin{lemma}  \label{lemu}  Let $A$ be an operator algebra
with a  r.c.a.i..  Then the canonical `left regular
representation' of $A$ on itself yields completely contractive
embeddings (i.e. 1-1 homomorphisms)
$$A \hookrightarrow \M_\ell(A) \hookrightarrow CB(A) \; \; ,$$
and the first of these embeddings, and their composition,
are completely isometric.
\end{lemma}
 
\vspace{4 mm}
 
We remark that the canonical
 embedding $\M_\ell(A) \hookrightarrow CB(A)$,
where $A$ is an operator algebra with r.c.a.i.,
 is not in general completely isometric, or even isometric
(see Example \ref{Exmdb}).   This has implications for our theory
of multipliers in \S 3.

 
\begin{lemma} \label{pid2} 
Suppose that $a \in B(H)$, and
$\{ e_\alpha \}$ is a net of contractions in $B(H) $ such that
$a e_\alpha \rightarrow a$.  Then
% is subalgebra of a $C^*$-algebra, and suppose that $J$ has
%a r.c.a.i. $\{ e_\alpha \}$.   Then
$a e_\alpha e_\alpha^* \rightarrow a$,
$a  e_\alpha^* e_\alpha \rightarrow a$, and
$a  e_\alpha^* \rightarrow a$.
\end{lemma}
 
%Alternatively, one may use:
%
%\begin{theorem} \label{bez} (\cite{BEZ})  A linear 
%$T : X \rightarrow X$ on an operator space is in 
%$Ball(\M_\ell(X))$ if and only if $T \oplus Id$ is a
%complete contraction $C_2(X) \rightarrow C_2(X)$.
%\end{theorem}
%Here $C_2(X)$ is `the first column of $M_2(X)$; that is
%$C_2(X) = M_{2,1}(X)$.  
%As we said above, $\M_\ell(X)$ is an operator algebra.  The
%matrix norms on $\M_\ell(X)$ may be described via the
%natural isomorphism $M_n(\M_\ell(X)) \cong \M_\ell(M_n(X))$.
%That is the norm of a matrix $[T_{ij}]$ of multipliers may be taken to
%be the norm in $\M_\ell(M_n(X))$ of the map
%$$[x_{ij}] \mapsto [\sum_k T_{ik}(x_{kj})] \; \; .$$
 
\section{The ideal envelope}

%We begin this section by stating a few simple
%lemmas from \cite{Bid1}  which will be useful
%later.  

We begin by sketching Hamana's construction of the injective and 
the triple envelope of operator spaces.  The reader who is not 
familiar with this will need to consult \cite{Ham2} for 
more details and notation; unfortunately
the material below will be a little technical for 
those not versed in `Hamana-theory'.   See also
the introductions of \cite{BPnew} and \cite{BShi}). 
Let $X\subset B(H)$ be 
an operator space, and consider the Paulsen 
operator system
$${\mathcal S}(X):=\left[
\begin{array}{cc}
\Co 1_H & X \\
X^* & \Co 1_H
\end{array}
\right]
\subset M_2(B(H)) \; .$$   
One then takes a minimal (with respect to a certain 
ordering) completely positive 
${\mathcal S}(X)$-projection $\Phi$ on
$M_2(B(H)).$  
%We say that 
%Im$ \;\Phi$ is an {\em injective envelope} of ${\mathcal S}(X)$.
By a well-known result of Choi and Effros,
Im$ \;\Phi$  is a $C^*$-algebra with the multiplication $\odot$ defined by
$\xi \odot \eta := \Phi(\xi \eta)$ for $\xi, \eta \in$ Im$\Phi,$ and the
other algebraic operations and norm are the usual ones.   
%There is a canonical 
%projection (namely the canonical idempotent in ${\mathcal S}(X)$
%having a 1 in the 1-1-corner and $0$ elsewhere)
%in this C*-algebra with respect to which the 
One may write   
\begin{center}
Im$ \; \Phi = \left[
\begin{array}{cc}
I_{11} & I(X) \\
I(X)^* & I_{22}
\end{array}
\right] \; \subset M_2(B(H)) ;$
\end{center}
and $I(X)$ is an {\em injective envelope}
of $X,$ and $I_{11},$ $I_{22}$ are 
injective C*-algebras.    We sometimes write 
$I_{11}$ as $I_{11}(X)$, say, to emphasize the dependence 
on $X$.  

We continue to think of 
%$Im \; \Phi \subset M_2(B(H))$, and
$I_{11},$ $I_{22},$ $I(X)$ and $I(X)^* $
as subsets of $B(H)$, however the operation
$\odot$ induces new products between elements of
$I_{11},$ $I_{22},$ $I(X)$ and $I(X)^*.$   To distinguish these
multiplications from the original product on $M_2(B(H))$ we write
the new products as $\circ$.  

By a well known trick one may also decompose
$$\Phi=\left[
\begin{array}{cc}
\psi_1 & \phi \\
\phi^* & \psi_2
\end{array}
\right].$$ 

Also one may write the $C^*$-subalgebra of $Im \; \Phi$ (with the 
new product) generated by 
\begin{center}
$\left[
\begin{array}{cc}
0 & X \\
0 & 0
\end{array}
\right]      $
\end{center}
as 
\begin{center}
$\left[
\begin{array}{cc}
{\mathcal E}(X) & {\mathcal T}(X) \\
{\mathcal T}(X)^* & {\mathcal F}(X)
\end{array}
\right]
\subset
\left[
\begin{array}{cc}
I_{11} & I(X) \\
I(X)^* & I_{22}
\end{array}
\right] \; .  $
\end{center} 
This defines $\T(X)$, it is clearly a
$C^*$-module or triple system (viewed
as the 1-2-corner of 
the $C^*$-subalgebra just introduced,
 its triple product is $x \circ y^* \circ z$).  
Indeed the span of
expressions of the form $a_1 \circ a_2^* \circ 
a_3 \circ  a_4^* \circ 
\cdots \circ a_{2n+1}$,
for $a_i \in X$, are dense in $\T(X)$.   
Thinking of 
$\T(X)$ as a triple system, and letting $i$ be the canonical 
map $X \rightarrow \T(X)$, we say that $(\T(X),i)$ is
%(a representative for)
a {\em triple envelope} of $X$.
                                        
 \begin{lemma}
\label{product}
Let A be a subalgebra of $B(H)$, and suppose that A has a r.c.a.i.
$\{ e_{\alpha} \}.$ Then
for all $a, b\in A$ we have (taking $X = A$ and using the notation above)
$$ab=\lim_{\alpha} a\circ
e_{\alpha}^*\circ b \; ,  \; \text{and} $$
$$\psi_1(a) = \lim_\alpha a \circ e_\alpha^*  \in I_{11} \; . $$
\end{lemma}
 
\begin{proof}  Using the notation above, and Lemma \ref{pid2},
we have 
$$\psi_1(a) = \lim_\alpha \psi_1(a e_\alpha^*) = \lim_\alpha  a \circ e_\alpha^*
 \; , $$         
the last step by definition of the $\circ$ product.  Similarly                                          
$$ab=\phi(ab)=\phi(\lim_{\alpha} ae_{\alpha}^*b)
=\lim_{\alpha} a\circ e_{\alpha}^* \circ b . $$ 
\end{proof}  

We recall that an equivalent definition of $\M_\ell(X)$ given in 
\cite{BPnew} was as $IM_\ell(X) =
\{ y \in I_{11} : y \circ X \subset X \}$.
Using the last lemma we see that if $A$ is an operator algebra
with r.c.a.i., then $\psi_1(a) \in IM_\ell(A)$ for all 
$a \in A$.  Thus if $j$ is $\psi_1$ restricted to $A$,
then $j : A \rightarrow IM_\ell(A)$.  By the last lemma,
$j(a) \circ b = ab$ for all $a,b  \in A$, so that $j$ corresponds 
to the left regular representation $\lambda : A
\rightarrow \M_\ell(A) \subset CB(A)$.         


\begin{proposition} \label{notfi}  Let $A$ be an operator 
algebra with r.c.a.i..
\begin{itemize}
\item [(1)] Under the same assumptions as  Lemma
\ref{product}, and notation
as above,
$$IM_{\ell}(A)=\{T\in IM_{\ell}(A) :
 T \circ j(A) \subset j(A) \}.$$
\item [(2)]  Elements of
$\M_\ell(A)$, considered as maps on $A$,
are right $A$-module maps.  That is, $\M_\ell(A) \subset CB_A(A)$ as
sets.   Also, $\M_r(A) \subset \; _ACB(A)$ as sets. 
\end{itemize}  
\end{proposition}
 
\begin{proof}
(1) Let $a \in A$, then         
$T\circ j(a)
= \lim_\alpha T\circ a\circ e_{\alpha}^*$,
which equals $j(T \circ a)$
if $T \in IM_{\ell}(A)$.  Conversely,
if $T\circ j(a) = j(a') $ for an $a' \in A$, 
then by the last formula and \ref{product} we
have $T\circ j(a) \circ e_\beta =  T\circ a e_\beta
= a' e_\beta$, so that $T\circ a = a' \in A$.   
%(b))=T\circ (ab)=T\circ (\lim_{\alpha \to \infty}a\circ
%e_{\alpha}^*\circ b)
%=\lim_{\alpha \to \infty} T\circ (a\circ e_{\alpha}^* \circ b)=\lim_{\alpha
%\to \infty} (T\circ a)\circ e_{\alpha}^* \circ b
%=(T\circ a)b=\lambda (T\circ a)(b)$

(2)  
%The first part
% follows from the line above the proposition, and 
%$$T \circ (ab) = T \circ j(a) \circ b =
%\sigma(T \circ a) \circ b = (T \circ a)b$.   
For the last assertion, if $T \in \M_r(A)$
then there exists $y \in I_{22}$ such that 
$A \circ y \subset A$, and $T$ is the
map $T(a) = a \circ y$.  Hence for $c \in A$,  
$$T(ca) = (ca) \circ y = \lim_\alpha c \circ e_{\alpha}^*\circ
a \circ y =  c T(a) \; \; ,$$
using \ref{product} twice.  The other is similar.
\end{proof}

\vspace{4 mm}

%Note that it was not at all clear in the proof of
%\ref{comu} whether $\{ T \in \M_\ell(A) : T \lambda(A) \subset
%\lambda(A) \}$ was actually equal to $\M_\ell(A)$.
%The previous result asserts 
%that it is, which is of great importance in \S 5.                  
In the statement of the following theorem, and in 
keeping with the usual presentation of Hamana theory
(in this case the triple envelope), we forget the 
original product on $B(H)$ mentioned in the above discussion.
That is, the products in the  statement below are  
the $\circ$ operation in the above discussion.   However 
in the {\em proof} we go back to the $\circ$ notation.  
 
\begin{theorem} \label{Thlem}
Let $A$ be an operator algebra with r.c.a.i. $\{ e_\alpha \}$.
%Under the same assumption of Lemma \ref{product}, t
Suppose that $(\mathcal{T}(A),i)$ is a triple envelope 
for $A$, and let $\mathcal{E}(A) = \mathcal{T}(A) 
\mathcal{T}(A)^*$.   The map
$\psi : \mathcal{T}(A)\longrightarrow 
\mathcal{E}(A)$  defined by
$\psi (x):=\lim_{\alpha} x i(e_{\alpha})^*$ is a 
well defined complete
isometry, and $\psi \circ i$ is a homomorphism $j$ on A.   The range of 
$\psi$ is a left ideal $J$ of the $C^*$-algebra
$\mathcal{E} (A)$, and $\psi$ is also a
triple morphism onto $J$.   Thus $(J,j)$ is another triple 
envelope of $A$.
%this left ideal $J$, with $j$ a
%completely isometric homomorphism of $A$ onto 
%a subalgebra of the left ideal $J$ of ${\mathcal E} (A).$
Moreover $J J^* = {\mathcal E} (A),$ and any 
r.c.a.i for $A$ is taken by $j$ to a r.c.a.i. 
for $J$.   Finally, $\psi(x) \psi(y)^*
= x y^*$ for any $x, y \in \mathcal{T}(A)$.
\end{theorem}
 
\begin{proof}  By `abstract nonsense' we can assume that 
$(\mathcal{T}(A),i)$ is the triple envelope considered 
above \ref{product} (in short because
the statement we are attempting
to prove is invariant under the notion we called 
$A$-isomorphism in \cite{BShi} Appendix A). 
We use the notation of \ref{product} and above \ref{product}.
Since elements spanning a dense subset of 
$\mathcal{T}(A)$ are $\circ$ products ending with a term in $A$,
it follows from Lemma \ref{product} that $
\lim_{\alpha} x \circ e_{\alpha}^*$ exists in $\mathcal{E}(A)$
for any $x \in \mathcal{T}(A)$.  Clearly 
$\psi$ is a complete contraction extending the map $j$ 
introduced above \ref{notfi}.
If $x\in A,$ then $$\| \psi (x)\| =\lim_{\alpha} \|
x\circ e_{\alpha}^*\| \geq \lim_{\alpha} \|
x\circ e_{\alpha}^* \circ e_{\beta} \|
= \|x e_{\beta} \|$$
using \ref{product}.  Taking the limit shows that 
$\psi$ is isometric on $A$.  A similar argument using 
associativity of $\circ$ shows that $\psi$ is isometric on $\mathcal{T}(A)$.
Similarly, $\psi$ is a complete isometry.
If $a, b\in A,$ then 
$$\psi(ab)=\lim_{\alpha} (ab)\circ
e_{\alpha}^*
=\lim_{\alpha}(\lim_{\beta}a\circ e_{\beta}^*\circ
b)\circ e_{\alpha}^*
=(\lim_{\beta}a\circ e_{\beta}^*)\circ (\lim_{\alpha}
b \circ e_{\alpha}^*)=\psi (a)\circ \psi (b).$$
 Note also that
$$\psi(a) \circ \psi(b)^* = 
\lim_{\alpha} \lim_{\beta} a\circ e_{\alpha}^* \circ 
e_{\beta} \circ b^* = \lim_{\beta} (ae_{\beta}) \circ b^*
= a \circ b^* \; \; .$$
Thus in the language of the statement of the 
theorem $\psi(i(a)) \psi(i(b))^* = i(a) i(b)^*$.   
 
By looking at the natural dense subsets of 
$\mathcal{T}(A)$ and $\mathcal{E}(A)$ it is easy 
to argue that the range of
$\psi$ is a left ideal $J$.  By a fact mentioned in the 
introduction this implies that $\psi$ is a triple 
morphism.   Since $j$ is the restriction of 
$\psi$ to $A$, the remaining 
assertions are easy.   One needs to use the fact that 
$\psi$ is a triple morphism, and that $\mathcal{T}(A)$
and therefore consequently $J$, have  dense subsets spanned by
terms which are alternating products as mentioned 
above \ref{product}. \end{proof}
 
\vspace{5mm} 
 
\begin{theorem} \label{Thlemi}  Let $A$ be an operator algebra
with a r.c.a.i., let $(I(A),i)$ be its injective envelope 
 discussed above \ref{product} say,
and let $B$ be the injective unital  
$C^*$-algebra $I_{11}$ (also discussed above).    Then there is an
orthogonal projection $e \in B$, such that $I'(A) = (B e,j)$ is also an
injective envelope for $A$, where $j$ is as above.
%in \ref{Thlem}.
% ($= \sigma = $ restriction of $\psi$ there).  
%injective envelope 
%$(B e,j)$ for $A$, identified with a 
%principle left ideal $B e$, for a projection
%$e \in B$, and via this identification 
For this new injective envelope, $I'_{22} \cong eBe$ and 
$I'_{11} \cong B$.  The completely isometric embedding
$j : A \rightarrow I'(A) \subset B$ is a
homomorphism.  Moreover if $A$ has a right identity $f$ then 
$j(f) = e$.
\end{theorem}  

Before we prove this, we remark that we may define the {\em ideal 
injective envelope} 
%$II(A)$
of an operator algebra $A$ with r.c.a.i.,
to be $I'(A)$ as in the theorem.  
%It is not hard to see
%that if $A$ is a left ideal $J$ in a 
%$C^*$-algebra, then the construction outlined before
%\ref{product} collapses considerably.  In particular,
%by Hamana's universal property of $\T(A)$ (\cite{Ham2} and the fact 
%that $J$ is a triple envelope of $J$, we see that t a  ADD HERE.
 
\vspace{4 mm}
 
\begin{proof}   
%This follows mostly from the previous result.  
%In the case that $A$ has a right contractive identity
%$f$, the map $\psi(x) = x \circ f^*$ defines a map
%$I(A) \rightarrow I_{11}$.  By Hamana's
%`essential property' for $I(A)$ \cite{Ham2},
%$\psi$ is a complete isometry.  That its range is 
%an ideal is clear: for $w \in I_{11}$ we have
%$w \circ \psi(x) = \psi(w \circ x)$, and one proceeds easily 
%from here.
%
%If $A$ only has a r.c.a.i., the above  idea
%does not quite 
%seem to work.  In this case, d
Define $\psi : I_{11} \circ A
\rightarrow I_{11}$ by $\psi(x) = \lim_\alpha
x \circ e_\alpha^*$.  By \ref{product} this limit exists and 
$\psi$ is clearly completely contractive.  Note that 
$\psi$ is also a left $I_{11}$-module map, so that by a
result in \cite{BPnew} we may extend $\psi$ to a
completely contractive left $I_{11}$-module map 
 $\hat{\psi} : I(A) \rightarrow I_{11}$.   The restriction 
of this map to $A$ was the map called $j$ above.
By Hamana's
`essential property' for $I(A)$ \cite{Ham2}, $\hat{\psi}$ is a 
complete isometry.  By \cite{BPnew} 2.7, there is
a $v \in Ball(I(A))$ such that $\hat{\psi}(x) = x \circ v^*$,
for all $x \in I(A)$.  Also its range is a left ideal of 
$I_{11}$, so that $\hat{\psi}$ is a triple morphism 
by a fact at the end of the introduction.

Next define a map 
$I_{11} \circ j(A) \subset I_{11} \rightarrow I(A)$ by
$z \mapsto \lim_\alpha z \circ e_\alpha$.   To see that 
this limit exists note that for $x \in I_{11}$ and $a \in A$
we have by the type of calculations found in 
the proof of \ref{Thlem}, that $\lim_\alpha x \circ j(a) \circ e_\alpha
= x \circ a$.  So the limit exists on $I_{11} \circ j(A)$, and moreover
this map is exactly the map $x \circ j(a) \mapsto 
x \circ a$.   It is thus a completely contractive left 
$I_{11}$-module map, and extends by \cite{BPnew} to a
completely contractive left $I_{11}$-module map
$\mu : I_{11} \rightarrow I(A)$.  There clearly exists
$w \in Ball(I(A))$ such that $\mu = - \circ w$.  Now notice 
that the composition $\mu \hat{\psi}$ is a completely 
contractive map $I(A) \rightarrow I(A)$, which restricts to the 
identity map on $A$.   By Hamana's rigidity property \cite{Ham2},
$\mu \hat{\psi} = Id$ on $I(A)$.  Thus $\hat{\psi}$ is a
complete isometry, $\mu$ is onto.  Moreover since 
$x \circ (v^* \circ w) = x$ for all $x \in I(X)$, it follows 
from 
%(the `other-handed version' of)
 \cite{BPnew} Corollary 1.3, that $v^* \circ w$ is the 
identity of $I_{22}$.  Thus by a well know operator theory fact,
$v = w$.    We define $e = v \circ v^* \in I_{11} = B$, this 
is an orthogonal projection.   Moreover $Ran \; \hat{\psi} =
B e$, as may be seen easily from the above.  It follows 
immediately that  $(B e,j)$ is an 
injective envelope for $A$, where $j(a) = a \circ v^*$ (which coincides
with the map $j$ of previous results), which is a 
homomorphism.  

Note that by \cite{BPnew} Theorem 1.8, $I'_{11} \cong M(I'(X) I'(X)^*)
= M(I(X) I(X)^*) \cong I_{11}$, and $I'_{22} \cong M(eBe) = eBe$. 

Finally, if $A$ has right identity $f$ then we may take 
$v$ above to be $f$, so that $j(f) = f \circ f = e$.   
\end{proof} 
 
\vspace{4 mm}

%If $A$ has a right contractive identity $e$ , then the map 
%$x \mapsto e \circ x \circ e^*$ (resp. $x \mapsto e \circ x $)
%is easily shown to
%be a 1-1 *-homomrphism $I_{22} \rightarrow I_{11}$
%(resp. 1-1 triple morphism $I_{22} \rightarrow I(A)$).  
%This is no doubt true in the case of a r.c.a.i. too.
%However we do not ia 
%\begin{theorem} \label{Thlem}  Let $A$ be an operator algebra
%with a right contractive approximate identity.
%% (I think this
%%is OK, certainly my proof is OK if $A$ has 
%%right identity $e$ of norm 1.
%%Then the $C^*$-algebra
%%$\E(A)$ is unital (i.e. $A$ is left e.n.v. in the language of
%%\cite{BShi}), and
%Then $\T(A)$ may be
%identified with a left ideal of the $C^*$-algebra
%$\E(A)$, and moreover the `Shilov embedding'
%$j : A \rightarrow \T(A)$ may be taken to be a
%homomorphism onto a subalgebra of the left ideal
%$\T(A)$ of $\E(A)$, and $j$ takes the 
%r.c.a.i. of $A$ to  
%%with the property that $j(e)$ is
%a r.c.a.i. for $\T(A)$.
%\end{theorem}
% 
%\begin{proof}  Suppose that $A$ is a subalgebra of $B(H)$ say.
%%, then if $A$ has right identity $e$ of norm 1 then
%%$e$ is an orthogonal projection in $B(H)$.
%Consider the $C^*$-subalgebras $C^*(AA^*)$ and
%$C^*(A^* A)$, and the TRO $\He$ generated
%by $A$, inside $B(H)$.   Then $\He$ is a strong Morita equivalence
%$C^*(AA^*)-C^*(A^* A)$-bimodule.   Moreover if 
%$A$ has a r.c.a.i. $\{ e_\alpha \}$ and $z \in A$ then 
%$z e_\alpha^* \in C^*(AA^*)$, but 
%$z e_\alpha^* \rightarrow z$ by  \ref{pid2}, so that 
%$A \subset C^*(AA^*)$.  Hence also $\He \subset C^*(AA^*)$.
%%right identity $e$ of norm 1, then $\He = \He e = \He e^* \subset
%%C^*(AA^*)$, and a similar argument should work in the general case 
%%using \ref{pid2}.  Indeed in one or two other 
%%places below we assume the existence of $e$, 
%%I will check, and it should be easy to do,
%% the general case using \ref{pid2}.   
%
%Now it is clear that $\He$ is a subalgebra,
%and a left ideal of $C^*(AA^*)$, and therefore a
%sub-TRO of $C^*(AA^*)$.   Clearly the r.c.a.i. for $A$ is a 
%r.c.a.i. 
%$e$ is a right identity 
%for $\He$.   Now let $W$ be the
%Shilov boundary submodule of $\He$.   By Hamana 
%theory (\cite{Ham2} or \cite{BShi} A.12), we
%may take $\T(A)$ to equal $\He/W$, and $\E(A)$ will then 
%be the $C^*$-algebra $(\He/W)(\He/W)^*$.  By the basic theory of
%quotient Morita contexts \cite{RieffelUn}, this last 
%$C^*$-algebra is the same as
%%$\He/W$ is a a left
%$C^*(AA^*)/(W W^*)$.
%%-module which is a strong Morita equivalence bimodule for this algebra.  .
%That is, $\E(A) \cong C^*(AA^*)/(W W^*)$.   
%Consider the natural maps (most of which we've
%just described)
%$$ A \hookrightarrow \He 
%\hookrightarrow C^*(AA^*) \rightarrow C^*(AA^*)/(W W^*) \cong \E(A).$$
%These maps are all homomorphisms quite clearly, and all but the first
%are *-homomorphisms or triple morphisms.   The ensuing map
%$\He \rightarrow C^*(AA^*)/(W W^*)$ is a triple morphism,
%let $K$ be its kernel.
%% which has kernel which contains $W$, since
%Now $W W^* W = W$ as we stated in the introduction 
%for a TRO (or $C^*$-module \cite{La,W-O}) 
%so that if $w \in W$ then using \ref{pid2} we have
%$w = \lim w e_\alpha^* \subset W W^* W e_\alpha^*  \subset W W^*$.   
%Conversely, for any $x \in K$ 
%we have $x \in W W^*$, so that $x e_\alpha \in
%W W^* e_\alpha \subset W$ (since $W$ is a bimodule).  Thus in the limit
%$x \in W$.  Hence the kernel $K$ is
%$W$, and therefore by the basic theory of triple morphisms
%\cite{Ham2} we obtain a completely isometric triple morphism 
%$\He / W \hookrightarrow C^*(AA^*)/(W W^*) \cong \E(A)$.  This embedding 
%has range which is a left ideal as is easily checked.
%We saw above that 
%we may take $\T(A)$ to equal
%$\He / W$, and thus $\T(A)$ may be identified with the left ideal
%$\He/(W W^*)$ of the $C^*$-algebra
%$C^*(AA^*)/(W W^*)$.   Thus
%we may view the `Shilov embedding'
%$j : A \rightarrow \T(A)$ to be $j(a) = a + W W^* \in C^*(AA^*)/(W W^*)$.
%But then $j$ is clearly a homomorphism, with $j(e_\alpha)$ the r.c.a.i.
%of the left ideal $\He/(W W^*)$.
%\end{proof}                                   

%\vspace{4 mm}

It is clear from \ref{Thlem}
that for an operator algebra $A$ with r.c.a.i., $\T(A)$ 
may be taken to be a left ideal in a $C^*$-algebra, and this
left ideal therefore  
possesses  Hamana's universal property of the triple 
envelope (see \cite{Ham2}
or \cite{BShi} Appendix A).  However examining this property in 
this case gives an interesting refinement, by viewing 
$(\T(A),j)$ as an {\em left ideal extension} of $A$.   

To be 
more specific we say that a pair $(J,i)$ consisting 
of a left ideal $J$ in a $C^*$-algebra, and a completely isometric
homomorphism $i : A \rightarrow J$, is 
a {\em left ideal extension} of $A$
if $i(A)$ `generates $J$ as a triple system'.  That is, the span of 
expressions of the form $i(a_1) i(a_2)^* i(a_3)  i(a_4)^* 
\cdots i(a_{2n+1})$,
for $a_i \in A$, are dense in $J$.   It follows from this
that $\{ i(e_\alpha) \}$ is a r.c.a.i. for $J$ if 
$\{ e_\alpha \}$ is a r.c.a.i. for $A$.  Then it is clear from 
\ref{Thlem} that $(\T(A),j)$ may be taken to 
be a left ideal extension of $A$, 
and it is the `minimal such'.   The new point in the theorem below
is that $\tau$ may be chosen to be a homomorphism:

\begin{corollary} \label{upie}  Let $A$ be an 
operator algebra with r.c.a.i..  Then there 
exists a left ideal extension $({\mathfrak J}_e(A),j)$ of $A$, 
with ${\mathfrak J}_e(A)$ a left ideal in a $C^*-$algebra
$\E(A) = {\mathfrak J}_e(A) {\mathfrak J}_e(A)^*$, such that for 
any other left ideal extension $(J,i)$ of $A$, there exists 
a (necessarily unique and surjective)
completely contractive homomorphism $\tau: J 
\rightarrow {\mathfrak J}_e(A)$, which is also a
triple morphism, such that $\tau \circ i = j$.  Thus 
${\mathfrak J}_e(A)/(ker \; \tau)  \cong J$ completely isometrically
homomorphically (i.e as operator algebras) too.
Moreover $({\mathfrak J}_e(A),j)$ is unique in the following sense:
given any other $(J',j')$ with this universal property,
then there exists a surjective completely isometric 
homomorphism $\theta : {\mathfrak J}_e(A) \rightarrow J'$ such that 
$\theta \circ j = j'$.
\end{corollary}

\begin{proof}  The uniqueness is fairly obvious and 
standard.  The existence follows from 
\ref{Thlem} (setting ${\mathfrak J}_e(A) = (J,j)$ there),
together with Hamana's universal property 
for the `triple envelope' mentioned above. Indeed we have:
 $$\tau(x y) = \lim \tau(x i(e_\alpha)^* y)
= \lim \tau(x) j(e_\alpha)^* \tau(y) = \tau(x) \tau(y)$$
for $x, y \in J$, using    
Lemma \ref{pid2} and the observations above.
\end{proof}

\vspace{4 mm}

We call $({\mathfrak J}_e(A),j)$ the {\em left ideal envelope} of 
$A$, and continue to write $\E(A) = {\mathfrak J}_e(A) {\mathfrak J}_e(A)^*$.
Again, $j$ will be called the {\em Shilov embedding
homomorphism}.

%
%\begin{corollary} \label{liie}  Let $J$ be a left ideal 
%in a $C^*$-algebra.   Then 
%One can show \cite{Bid1} that the canonical embedding of 
%a left ideal $J$ 
%in a $C^*$-algebra, 
%into its ideal injective envelope $B e$ (see
%\ref{Thlemi}) is an ideal homomorphism.
%
%\begin{proof}   Using \ref{upie}, there exists a
%surjective triple morphism $\tau : J \rightarrow {\mathfrak J}_e(J)$,
%which is also a homomorphism, such that $\tau = j$.  
%Clearly $J_e(J)$ is a subalgebra and sub-TRO of 
%$\E(J)$, and consequently also of $B$ and $B e$.  
%\end{proof} 
 
%$J$ is a subalgebr  
%Let us prove and at the same time 
%amplify this principle.  So suppose that $A$ is an
%operator algebra with a r.c.a.i..     

\begin{corollary} \label{usf}  Let $A$ be an operator algebra
with r.c.a.i., and let $j : A \rightarrow \E(A)$ be the 
Shilov embedding homomorphism.
% constructed in the proof of \ref{Thlem}.
Then $\M_\ell(A) \cong \{ T \in LM(\E(A)) : T j(A) \subset j(A) \}$
completely isometrically isomorphically.  \end{corollary} 
 
\vspace{4 mm}
      
This last corollary above
is a restatement of \ref{notfi}, using
the definition of $\M_\ell(A)$ given in the introduction.
%\cite{BShi} \S 4.
 
\section{The left multiplier algebra of an algebra with right 
approximate identity}

The readers first thought here might be that this case must be
analogous to
%treatable in a similar way to 
the case of the left multiplier algebra
of an operator algebra with 
l.c.a.i., which was treated in \cite{Bid1}.
However in fact the theory is rather different.
Part of this may be seen by considering the basic example
of $A = C_n$ (the $n \times n$ matrices
`supported on' the first column).  
It is fairly clear that
its left multiplier operator algebra `should be'
$M_n$.

After a little thought about operator algebras 
with r.c.a.i., one comes up with the following list
(which we shall refer to throughout this section) of
 `possible candidate definitions' for $LM(A)$ in this
case (all of which contain $A$ completely 
isometrically):   
\begin{enumerate}
\item [(I)] $\{ x \in A^{**} : x \hat{A} \subset \hat{A} \}$,
\item [(II)] $\{ T \in B(H) : T \pi(A) \subset \pi(A) \}$, where
$\pi : A \rightarrow B(H)$ is a  
completely isometric nondegenerate representation,
\item [(III)] $\M_\ell(A)$, 
\item [(IV)] $CB_A(A) \; \; \; \; $ (right module maps), 
\item [(V)] the `maximum essential left multiplier extension of $A$'.
\end{enumerate}
We will spell out later what is meant in this setting by 
% `nondegenerate representation' and 
 `essential left multiplier extension'.
One can rule out (I) fairly quickly as a plausible candidate
since it is not unital (e.g. take $A = C_n$).
However one would hope that most of the other four items are
completely isometrically isomorphic.   Unfortunately, most of the
equivalences amongst these items 
break down if $A$ is a general operator algebra
with r.c.a.i..
% (although at least one has that 
%$A \subset LM(A)$ in this case, unlike in  \cite{Bid1} \S 4).  
Nonetheless one of these five candidates will
emerge from our study below as the `winner', namely as the
appropriate version of
$LM(A)$ in this case.  This seems quite interesting.
              
In the special case of a  
left ideal $J$ in a $C^*$-algebra things are much better.
Most of these five items  are then
 completely isometrically isomorphic.    Indeed
in this case we shall see that most of these coincide with the 
usual left multiplier algebra
$LM(J J^*)$ of the $C^*$-algebra $J J^*$.  
One may view this as being predicted by the theory
of Hilbert $C^*$-modules.  See also \cite{KP}.  Indeed 
the analogous `left multiplier $C^*$-algebra' $LM^*(J)$,
which may be defined to be $M(J J^*)$,
or equivalently $A_l(J)$ in the notation of \cite{BShi},
also has a very satisfactory theory which has 
essentially been done in \cite{KP}.  Hence we 
shall not mention $LM^*(J)$ again here, leaving any details
of its theory to the interested reader.
  
In this special case of a left ideal $J$ in a $C^*$-algebra
we will take  the term 
`essential left multiplier extension'  used 
in (V), to be defined as in
the proof of (4) in the main theorem 
in \cite{Bid1} \S 4.  Of particular 
interest, in (II),
are the {\em ideal representations} of $J$
discussed in \cite{Bid1}, that is,
the completely isometric representations $\pi : J
\rightarrow B(H)$ which 
are restrictions of faithful 
*-representations $J J^* \rightarrow B(H)$ (or equivalently,
the completely isometric representations of $J$ which are also 
triple morphisms). 
 
%\begin{proof}    .
%Consider the following conditions:
%\begin{itemize}
%\item [(i)] $\pi$ is nondegenerate (i.e. $\pi(A) H$ is dense in $H$);
%\item [(ii)]  $\pi$ is a triple morphism;
%\item [(iii)]  $\pi(A) \pi(A)^* H$ is dense in $H$.
%\end{itemize}
%Then (iii) implies (i), and (i) and (ii) together implies (iii).
%\end{lemma}
 
 
%\begin{proof}  Clearly (iii) implies (i).  If $\pi$ is a triple morphism
%then $\pi(A)$ is a TRO, and $\pi(A) \pi(A)^* H$ contains 
%$\pi(A) \pi(A)^* \pi(A) H = \pi(A) H$.
%%$, which contains $H$ $ is a $C^*$-algebra,
%%and there is an induced *-homomorphism from $J J^*$ onto
%% $ \pi(A) \pi(A)^*$.   
%\end{proof}

\begin{theorem} \label{lmli}  Let $J$ be a left ideal of a 
$C^*$-algebra.   The algebras in (III)-(V) are all
completely isometrically isomorphic to
$LM(J J^*)$.   Any algebra in (II)-(V) contain
 the algebra in (I) completely isometrically isomorphically
as a (proper, in general) subalgebra.  If, further,
 in (II) we only consider 1-1 nondegenerate ideal representations 
$\pi$, 
then the algebras in (II)-(V) are all
completely isometrically isomorphic.
 Also $B_J(J) = CB_J(J)$ isometrically. 
\end{theorem}

\begin{proof}  In this 
case $J$ is a $C^*$-module, and $\M_\ell(J)$ may easily be seen, from
\cite{BShi} A.4 if necessary, to be
$LM(J J^*)$.   Clearly $CB_J(J) \subset CB_{J^*J}(J)$, and conversely,
if $T \in CB_{J^*J}(J)$ then for $x, y \in J$ we have
using Lemma \ref{pid2}
that 
$T(xy) = \lim T(x e^*_\alpha y) = \lim T(x) e^*_\alpha y
= T(x) y$, if $\{ e_\alpha \}$ is the r.c.a.i. in 
$J$.   Thus $CB_J(J) = CB_{J^*J}(J)$.
By a theorem of Lin (\cite{Lin} 1.4)
$B_{J^*J}(J) =  LM(J J^*)$, and the operator space version of 
this is true too (see \cite{Bna}), that is
 $CB_{J^*J}(J) = LM(J J^*)$ completely isometrically.
Thus (III) = (IV), and we obtain 
the last assertion of the statement of the theorem too.   
Looking at the definition 
of a
%Defining 
`essential left multiplier extension', and the associated `ordering', below,
or defining these as we did in 
 the proof of (4) in the main theorem
in \cite{Bid1} \S 4, it 
is now clear that (IV) is an essential left multiplier extension,
so that it is the  
biggest such.   That is, (IV) = (V).   
%It is easy to see that (I) is 
%too small, and is not even
%necessarily unital (e.g. $A = C_n$), nonetheless the algebra
%in (I) is completely isometrically isomorphic to a subalgebra
%of (IV) evidently, by a simple argument (the idea was seen in  
%the proof of (3) = (7)  in the main theorem
%in \cite{Bid1} \S 4).

We will defer until \ref{lmrci} the proof that an algebra in (II)
contains the algebra in (I) completely isometrically.
Finally, 
%it is easy to see that the algebra in (II) is an 
%essential left multiplier extension of $J$, and so is 
%contained in $CB_J(J)$.   Conversely, 
%recall that 
%$CB_J(J) = LM(J J^*)$ as we saw above.   G
given a nondegenerate
faithful *-representation $\theta : J J^* \rightarrow B(H)$, 
we know from the multiplier theory for $C^*$-algebras (see
e.g. \cite{Ped} 3.1.12)
that 
$LM(J J^*) = \{ T \in B(H) : T \theta(J J^*) 
\subset \theta(J J^*)$.   However since for any ideal 
$J$ we have $J = J J^* J$, if  
$T \in B(H)$, then  $T \theta(J) \subset \theta(J)$
 if and only if  $T \theta(J J^*) \subset \theta(J J^*)$.
Thus the algebra in (II) equals $LM(J J^*)$ too in this
case.
%$, which contains $H$ $ is a $C^*$-algebra,
%and there is an induced *-homomorphism from $J J^*$ onto
% $ \pi(A) \pi(A)^*$.   
\end{proof} 

\vspace{4 mm}
 
We have now completed our discussion of $LM(J)$ for a left ideal 
$J$ in a $C^*$-algebra, and turn to the 
%.  Henceforth we restrict ourself to the 
more general case of an arbitrary operator algebra 
$A$ with r.c.a.i..   Since this 
case is the most complicated, we will be a little more formal
and rigorous in our presentation below.   

We begin by defining a {\em  left multiplier
extension} of $A$
%  a little differently.  Indeed we will define
%it now to
to be a pair $(B,\pi)$ consisting of an operator algebra
$B$ with an identity of norm 1, and a completely isometric
homomorphism $\pi : A \rightarrow B$ such that $\pi(A)$ is a
left ideal of $B$.   We say that $(B,\pi)$
is an {\em essential left multiplier extension} of $A$
if in addition the canonical map
$B \rightarrow B(A)$ is 1-1.     Note that this canonical map
is then a completely contractive homomorphism into
$CB_A(A)$ (viewing $A$ as a right $A$-module).  In the
2-sided case $CB_A(A)$ is itself a left multiplier
extension of $A$, and is therefore the largest
left multiplier extension, but this is not true in general for us
(see example \ref{Exma}).   We next define an ordering
on essential left multiplier
extensions of $A$: namely $(B,\pi) \leq (B',\pi')$ if there
exists a (necessarily unique and one-to-one)
completely contractive homomorphism $\theta :
B \rightarrow B'$ such that $\theta \circ \pi = \pi'$.
We remark that in the theory in  \cite{Bid1} \S 4 or for
left ideals in $C^*$-algebras one may insist that
$\theta$ is completely isometric, but we are not certain
if we can make this requirement in general.  Note that this ordering
says that $B$ may be viewed as a subalgebra of $B'$, but
with a possibly bigger norm.
 
%We remark in passing that a
% 2-sided unitization of $A$, such as the `multiplier unitization 
%mentioned after \ref{comu}, 
%should be an example of an
% essential left multiplier extensions.  It would be interesting
%to see if there is a `most useful' or `in some 
%sense unique' such unitization.
 
We will use the fact that
an essential left multiplier extension $(B,\pi)$
has the following rigidity property: if
$\theta : B \rightarrow B$ is a completely contractive
homomorphism extending the identity mapping
on $\pi(A)$, then $\theta$ is the identity mapping
(To see this note that
$\theta(b) \pi(a) = \theta(b\pi(a)) = b\pi(a)$,
so that $\theta(b) - b$ is in the kernel of the canonical
map $B \rightarrow CB(A)$).                                

We say that two left multiplier extensions
$(B,\pi)$ and $(B',\pi')$ are
{\em $A$-equivalent} if there
exists a unital completely isometric isomorphism $\theta :
B \rightarrow B'$  with $\theta \circ \pi = \pi'$.  Note that
this is an equivalence relation,
and that  `$\leq$' induces a well defined ordering on the
equivalence classes.   It follows that if
there exists  a maximum essential
 left multiplier extension of $A$, then it is
unique up to $A$-equivalence.
We remark further that if
two left multiplier extensions are $A$-equivalent,
and if one is essential, then so is the other.  We leave
the proof of this as an easy exercise.       

One more definition: we say that a representation $\pi : A
\rightarrow B(H)$ is a completely 
isometric {\em Shilov representation}, if 
$\pi$ is the restriction of a 1-1
*-homomorphism $\E(A) \rightarrow B(H)$
(viewing $A \subset \E(A)$ as in 
%the proof of \ref{Thlem} or 
\ref{upie} say
via the completely isometric homomorphism 
$j$ there).   Note that the `ideal representations'
 $\pi$ considered 
in Theorem \ref{lmli} happen also to be Shilov 
representations.
 
\begin{theorem} \label{lmrci}  Let  $A$ be an operator algebra
with r.c.a.i..   Then $\M_\ell(A)$ is the (unique up to
$A$-equivalence) maximum 
essential left multiplier extension of $A$.
That is, the algebras in items (III) and (V) 
in the list at the beginning of \S 5 are completely 
isometrically isomorphic.  
%We also have 
%$\M_\ell(A) \cong \{ x \in LM({\mathfrak J}_e(A) {\mathfrak J}_e(A)^*) :
%x j(A) \subset j(A) \}$ completely isometrically,  in the language of
%\ref{upie} say.  
Also $\M_\ell(A)$  
%contains the algebra in item
%(I) completely isometrically isomorphically
%as subalgebras.  It also 
contains, and dominates in the ordering above, the 
essential left multiplier extensions described in (II),
which in turn contain the algebra in 
(I) completely isometrically isomorphically.
However if $\pi$ in (II) is a Shilov representation of $A$
then 
the algebra in (II) is completely isometrically isomorphic to
(III) and (V).
The algebra in (IV), namely $CB_A(A)$,
is not in general an operator algebra, and its subalgebra
which corresponds to 
the algebra in (III) (and (V)) does not correspond isometrically.
That is, the canonical 1-1 homomorphic embedding of $\M_\ell(A)$ into 
$CB_A(A)$ (or even $B_A(A)$) need not be 
isometric.
\end{theorem}
 

\begin{proof}   
That the canonical inclusion of (III) in (IV) is not isometric
may be seen in Example \ref{Exmdb}.   That $CB_A(A)$ is 
not in general an operator algebra may be seen in Example \ref{Exma}.

We have seen in \ref{notfi} that
$(\M_\ell(A),\lambda)$
is an essential left multiplier extension.
It is evident from the fundamental
properties of $\M_\ell(A)$ (from 
Theorem 4.10 (1) of \cite{BShi} say)
%, or from 
%the `multiplication theorem' from \cite{BPnew}, 
%or from the characterization 
%of multipliers in \cite{BEZ}, 
that given any essential
left multiplier extension $(B,\pi)$ of $A$, there is a
canonical completely contractive homomorphism
$\sigma : B \rightarrow \M_\ell(A)$.  It is obvious  that
via this homomorphism $(B,\pi) \leq (\M_\ell(A),\lambda)$. 
So $(\M_\ell(A),\lambda)$ is the maximum essential
left multiplier extension.  
%If in addition $(B,\pi)$ is a maximum essential
%left multiplier extension, then  there            
%exists a completely contractive homomorphism
%$\sigma' : \M_\ell(A) \rightarrow B$ with
%$\sigma' \circ \lambda = \pi$.  Then by the rigidity
%property mentioned above the statement of the proposition
% we have
%$\sigma' \circ \sigma = Id$ and $\sigma \circ \sigma'
%= Id$.  Thus $(B,j)$ is $A$-equivalent to
%$(\M_\ell(A),\lambda)$.  This proves the uniqueness.

If  $\pi : A \rightarrow B(H)$ is a
completely isometric
`nondegenerate representation' (by which we mean at least
that $\pi(A) H$ is dense in $H$), then
$\{ T \in B(H) : T \pi(A) \subset \pi(A) \}$ is fairly 
evidently an essential
left multiplier extension of $A$, and hence is dominated
by $\M_\ell(A)$ (by the fact in the previous paragraph).  

%Consider item (I).  Let $j : A \rightarrow 
%\E(A)$ be the Shilov embedding homomorphism
%of \ref{Thlem} or
%\ref{upie},
%which is a completely isometric homomorphism such that 
%$j(A) j(A)^*$ generates $\E(A)$ as a $C^*$-algebra.
%We claim that $j^{**} : A^{**} \rightarrow   \E(A)^{**}$
%restricts to the desired completely isometric 
%isomorphism from the algebra in (I) into $LM(\E(A))$
%viewed as the appropriate subalgebra of $\E(A)^{**}$,
%and in fact into $\M_\ell(A)$ viewed as a subalgebra of 
%$LM(\E(A))$ as in \ref{usf}.
%%the definition of $\M_\ell(A)$ given in the introduction.  
%Certainly $j^{**}$ is a completely
%isometric homomorphism.
%% in the list at the beginning of \S 5
%%of candidates for the multiplier algebra,
%%we remark further that
% $\{ 
%If $x \in A^{**}$  is such that $x \hat{A} \subset
%\hat{A}$,
%%  is completely isometrically isomorphic to a
%subalgebra of $LM(A)$.   To see this, pick $x$ in
%this set, 
%and view $A$ as a subset
%of $\E(A)$ as in \ref{Thlem}, and view
%$A^{**} \subset \E(A)^{**}$.
%then it is apparent that $j^{**}(x) \in LM(\E(A))$, 
%since as we said above, 
%$j(A) j(A)^*$ generates $\E(A)$ as a $C^*$-algebra.
%Indeed clearly $j^{**}(x) j^{**}(\hat{A}) 
%= j^{**}(x \hat{A}) \subset j^{**}(\hat{A})$, which 
%means that indeed
%$x \in \M_\ell(A)$, using the first definition of $\M_\ell(A)$ given
%in \S 4 of \cite{BShi}. 

We now show any algebra in
(II) contains (I) completely isometrically.  
If $\pi : A \rightarrow B(H)$ is any (not necessarily 
nondegenerate) completely isometric homomorphism,  
consider the following sequence of
completely contractive homomorphisms:
$$A \hookrightarrow A^{**} \overset{\pi^{**}}{\rightarrow}
B(H)^{**} \rightarrow B(H)$$
where the first and last maps here are the canonical
inclusion and projection.   Let $\sigma$ be the composition of the
last two maps, restricted to $\{ x \in A^{**} : x \hat{A}
\subset \hat{A} \}$.  Then $\sigma(\hat{a}) = \pi(a)$ for
$a \in A$, and so for $x$ as above and $a \in A$ we have
$$
\sigma(x) \pi(a) = \sigma(x \hat{a})  \in \sigma(\hat{A}) 
\subset \pi(A)
 \; . $$   Thus $\sigma$ maps into the left idealizer of $\pi(A)$ 
in $B(H)$.  
To see that  $\sigma$ is isometric note that 
$$\Vert \sigma(x) \Vert \geq \Vert \sigma(x) \pi(e_\alpha) \Vert 
= \Vert \sigma(x \hat{e_\alpha}) \Vert = 
\Vert \pi(x \hat{e_\alpha}) \Vert = \Vert x \hat{e_\alpha} \Vert \; .$$
Now $1$ is a weak* limit point of the $\{ \hat{e_\alpha} \}$, and using
the seperate weak* continuity of the product on 
$A^{**}$,  we see  that $\Vert \sigma(x) \Vert \geq \Vert x \Vert$.
Similarly  $\sigma$ is completely isometric. 

Finally we discuss the equivalence of (III) with 
(II) for Shilov representations $\pi$.
%; the idea
%is similar to the argument we  saw for the relation of (I)
%and (III) above.
Take $\pi : \E(A)
\rightarrow B(H)$ to be any nondegenerate 1-1
*-homomorphism, then it
is easy to see that
$\pi_{|_A}$ is a nondegenerate completely isometric
homomorphism.   By the basic theory of multipliers of
$C^*$-algebras we may view
$LM(\E(A)) = \{ T \in B(H) :  T \pi(\E(A))
\subset \pi(\E(A)) \}$.   From 
\ref{usf} we thus have  $\M_\ell(A)$ completely 
isometrically isomorphic to
 $$\{ T \in B(H) :  T \pi(\E(A))
\subset \pi(\E(A)), \; \text{and} \;
T \pi(A) \subset \pi(A) \} = 
\{  T \in B(H) :  T \pi(A) \subset \pi(A) \}$$
since $j(A) j(A)^*$ generates $\E(A)$ as a $C^*$-algebra.
\end{proof}

\vspace{4 mm}

%The last theorem 
%shows that is that it is appropriate to define the
%`left multiplier algebra' of $A$ to be $\M_\ell(A)$, and
%we may therefore
%use the more compelling notation $LM(A)$ for this algebra,
%or any other
%left multiplier extension which is $A$-equivalent to
%$\M_\ell(A)$.
%  The theorem also warns the reader that $LM(A)$ is not
%completely isometrically
%contained in $CB(A)$ or $B(A)$:      


%One last remark on candidate (II) for the multiplier algebra:
%for certain reasons it seems appropriate
%to consider a completely isometric  homomorphism
%$\pi : A \rightarrow B(H)$ to be `nondegenerate' if it 
%satisfies the 
%stronger condition that $C^*(\pi(A) \pi(A)^*)$ acts nondegenerately
%on $H$ in the usual sense.  This coincides with the usual 
%definition if $A$ has c.a.i.. We defer further discussion of 
%this matter to another place.  

\begin{proposition} \label{ifoP}
Let $A$ be an operator algebra with r.c.a.i..
Then $CB_A(A)$ is an operator algebra if and only if
$CB_A(A)$ is completely isometrically isomorphic to $\M_\ell(A)$.
\end{proposition}

\begin{proof}  The one direction is trivial since 
 $\M_\ell(A)$ is an operator algebra.
To see the other note that by \ref{lemu} and \ref{notfi}
we have
 $\lambda(A) \subset CB_A(A)$, and $\M_\ell(A)
\subset CB_A(A)$ although the $\M_\ell(A)$-norm 
is possibly larger than the $CB_A(A)$-norm.   
Thus if $CB_A(A)$ is an
operator algebra then $(CB_A(A),\lambda)$ is an essential
left multiplier extension of $A$
a 1-1 completely contractive
homomorphism $CB_A(A) \rightarrow \M_\ell(A)$ (by the
equivalence of (II) and (V) in \ref{lmrci}).
It is easy to see that this homomorphism restricts to
the identity map on $\M_\ell(A)$, whose inverse is
completely contractive as a map $\M_\ell(A)
\rightarrow CB_A(A)$.
\end{proof}

\vspace{3 mm}

Next we consider some examples:

\begin{example} \label{Exma} \end{example}
Let $A \subset M_3$ be the subalgebra supported on the 1-2, 1-3, 2-2, 
and 3-3 entries only:
% (i.e. zero elsewhere).  
$$ A \; = \; \left[ \begin{array}{ccl} 0 & * & * \\
0 & * & 0 \\ 0 & 0 & * \end{array} \right] \; \; .$$
 Then $A$ is an
operator algebra with right identity of norm 1.
One may easily compute ${\mathfrak J}_e(A) = \T(A) = I(A) =
M_{3,2}$, and $\E(A) = M_3$, from which it follows that $\M_\ell(A)$ is the 
subalgebra of $M_3$ spanned by $A$ and $I_3$.  Thus $\M_\ell(A)$ is 
5 dimensional.  Indeed in this example, $\M_\ell(A)$ coincides with  what 
we called the `multiplier unitization' of $A$.
On the other hand the algebra given by item (I) in 
the list at the beginning of \S 5, is $A$ 
again, which is 4 dimensional.  A tedious but completely elementary 
algebraic computation shows that $CB_A(A)$, item (IV) on the aforementioned
list, is 8 dimensional.  This shows by \ref{ifoP}  that 
$CB_A(A)$ cannot be completely isometrically isomorphic to an 
operator algebra (in contrast to the 2-sided c.a.i. case). 
%For we claim that for any operator algebra $A$ with r.c.a.i.
%then $CB_A(A)$ is an operator algebra if and only if 
   

\begin{example} \label{Exmdb} \end{example}   It is perhaps true in the 
previous example that $\M_\ell(A)$ is not contained isometrically in 
$CB(A)$, but this seems difficult to check.  Instead we generalize 
this example to an interesting example of an
operator algebra $A$ with right identity of norm 1,
for which $\M_\ell(A)$ is fairly 
clearly  not contained isometrically in
$CB(A)$ (or $B(A)$).  To do this we take a $C^*$-algebra
$B$ with no nontrivial 2-sided
ideals, and a closed subspace $X \subset B$
which generates $B$ as a triple system.  
 To be more concrete 
one could take $B = M_3$ and 
$X \subset M_3$ as in Example 4.4 of \cite{BShi}.  
Consider $A \subset M_3(B)$ the 
subalgebra supported on the 1-2, 1-3, 2-2,
and 3-3 entries only (i.e. zero elsewhere), which has scalar multiples 
of the identity operator in the 2-2 and 3-3 entries, and has 
operators from $X$ in the 1-2 and 1-3 entries.  If one computes the 
triple system generated by
 $A$ in $M_3(B)$, one sees that one obtains 
$M_{3,2}(B)$ as in the previous example.  As in that example one 
sees that $\E(A) = M_3(B)$ and $\M_\ell(A)$ is the subalgebra of $M_3(B)$
spanned by $A$ and matrices supported only in the 1-1 entry, and that 
entry may be anything in $\M_\ell(X)$.  Now consider one of these 
last matrices, $T$ say, and let $a$ be its one nonzero 
entry (in $\M_\ell(X)$).  
Viewing $T \in CB(A)$ as a left multiplication map, 
it is clear that its norm or `cb-norm' is the same as the 
norm or `cb-norm' of its 1-1 entry, as a left multiplication map on $X$.
But in Example 4.4 of \cite{BShi} we showed that neither of these numbers
equals $\Vert a \Vert = \Vert T \Vert$ in general.

\vspace{4 mm}

%\begin{example} \label{rowfc} \end{example}  
%Consider the algebra $A$ of Example \ref{rowf}, but now assume that
%the $B$ there is an injective *-subalgebra 
%of $N = B(H)$, and that $I_H \in B$.  For example, $B = B(H)$.  We saw 
%in \ref{rowf} that 
%$A$ has r.c.a.i., and $\R(A) = \K_\infty(B)$.  It follows from the 
%theorem in \S 4 that $RM(A) = RM(\K_\infty(B))$,
%which is well understood by $C^*$-algebraists.   Computing 
%$LM(A)$ is a little trickier.   We will do this by using the injective 
%envelope $I(\cdot)$.  View 
%$$\K_\infty(B) \subset A  \subset M_\infty(B) \subset 
%B(\ell^2 \otimes H) \; \; . $$
%We may then identify $I(\K_\infty(B))$ is the range of 
%a completely contractive projection $\Phi$ on 
%$B(\ell^2 \otimes H)$ as described before \ref{product}.
%By Proposition 2.8 in \cite{BPnew}, $\Phi$ is  completely positive
%and unital.  Since $\Phi$ fixes $\K_\infty(B)$, it follows from Choi's 
%lemma (see e.g. \cite{BPnew} Lemma 1.6), 
%that $\Phi$ is a $\K_\infty(B)$-bimodule map.
%By looking at matrix units in $\K_\infty(B)$, it follows that 
%$\Phi$ is the identity map.   Thus $I(\K_\infty(B)) =
%M_\infty(B)$.      
%first 
%showing that $I(\K_\infty(B)) = $  babb .


%\vspace{4 mm}


{\bf Remark.}   If $A$ is an operator algebra
with r.c.a.i., then there is a most important element of $A^{**}$, namely
the right identity $E$ there.  A very natural question arises as to whether
$E \in LM(A)$, and more particularly whether $E \hat{A} \subset \hat{A}$?
In fact this is not true in general.  To see this suppose that 
we have an operator algebra $A$ with r.c.a.i. but no l.c.a.i.,
which has property ($\R$) discussed 
in \cite{Bid1}.   If $
E \hat{A} \subset \hat{A}$ then in the notation of  
\ref{notfi} there is a corresponding element of 
$e \in IM_l(A)$ such that $e \circ a = E a$ for $a \in A$.  In particular
for $b$ in the algebra $\R(A)$ defined in
\cite{Bid1} \S 2, we have $e \circ b = b$.    If one chooses  
an $A$ with $I(\R(A)) = I(A)$  (such examples exist)
then by the `rigidity property' of 
the injective envelope \cite{Ham2}, it follows that $e \circ a
= a$ for all $a \in I(A)$.  Thus in particular, for $a \in A$ we have
$E a = a$, from which it follows by \cite{BonsallDuncan} 28.7 that 
$a$ has a l.c.a.i., which is a contradiction.    

 
\vspace{4 mm}
      
{\bf Remark.}  (Unitizations.)   Notice that any time 
one has a (essential) left multiplier extension $(B,i)$ 
of an operator algebra $A$ with r.c.a.i., we get a
unitization of $A$ defined as $i(A) + \Co 1_B$.  
Conversely, any unitization of $A$ should be a
left multiplier extension of $A$.  

Taking 
the multiplier extension $\M_\ell(A)$ here, let us 
call the corresponding unitization
the `Shilov unitization' $A^1$
of $A$; it will be the unitization with the 
smallest norm. 
Of course another 
universal essential left multiplier extension may be 
constructed by considering a left idealizer in the 
direct sum of all essential left multiplier extensions of 
$A$.  We obtain an associated unitization of $A$, this one 
with the largest norm.  Indeed this may also described 
by simply assigning $A^1$ the supremum of all operator algebra
structures on $A^1$ coinciding with the usual one on $A$.

We do not know
%It also seems tricky, in view of the fact that 
%$CB_A(A)$ is not an operator algebra
%(see examples above), to show there 
in general if there is a simple 
formula for the norm on the unitization in terms of the 
norm on $A$, as there is in the case of an algebra
with two-sided c.a.i..   Of course if the 
span if $Id_X$ and $\lambda(A)$ inside $CB_A(A)$ is
an operator algebra, then this coincides  with 
the `Shilov unitization' mentioned above, by an argument
similar to \ref{ifoP}.
These matters deserve investigation.

\vspace{5 mm}

Finally, we end with some remarks on the 2-sided 
multiplier algebra
$M(A)$ of an operator algebra with l.c.a.i. say.   One natural 
candidate for $M(A)$ might be constructed as follows: take 
any candidate $(D,\mu)$ for $LM(A)$ (as in \cite{Bid1} \S 4), and define 
$M(A) = \{ x \in D : \mu(A) x \subset \mu(A) \}$.  It is easy to 
see that if we take a $(D',\mu')$ which is 
$A$-isomorphic to $(D,\mu)$, then one obtains the same $M(A)$,
up to completely isometric isomorphism, or up to a suitable 
notion of $A$-equivalence).  
Perhaps this
coincides with $M(\Li(A))$ if $A$ has property ($\Li$).  

One would hope that one might get the same 
algebra $M(A)$ by looking at $\{ x \in M_r(A) : x \rho(A) 
\subset \rho(A) \}$ where $\rho$ is the `right regular antirepresentation'
of $A$ on itself, but this is not correct
(consider $R_2$).  In fact this algebra is related to a `big' multiplier
algebra $BM(A)$ of $A$, 
defined as the `double idealizer'
of $j(A)$ in $M(B)$, where $B$ is the injective $C^*$-algebra
in \ref{Thlemi}.                                 
 Another approach to $M(A)$ is via
`double centralizers' as in \cite{PuR} and \cite{Pal} \S 1.2.
There must be some relations 
between all these candidates for $M(A)$, and some of the algebras
discussed at the end of  \cite{Bid1} \S 4 will also play a role.   The 
second author is currently investigating these matters
 to see if there are any satisfactory results here.  
 
\vspace{5 mm}
 
 
Acknowledgement:  Several results in this paper  will be included as   
part of the second author's Ph. D. thesis, done
under the direction of the first author during the '00-'01 academic 
year.   He expresses his gratitude to the first author for his kind 
instruction and advising.   

\begin{thebibliography}{99}
 
\bibitem{Arv1}  W. B. Arveson,
{\em Subalgebras of }$C^{*}-${\em algebras,}
 Acta Math. {\bf 123 }(1969), 141-224.

 
%\bibitem{Bhmo} D. P. Blecher,  
%{\em  A generalization of Hilbert modules,}
%J. Funct. An. {\bf 136} (1996), 365-421.    
\bibitem{Bnat}  D. P. Blecher,  {\em Some general theory of
operator algebras and their modules,} in
 {\em Operator algebras and
applications\/},
A. Katavalos (editor), NATO ASIC, Vol.
495, Kluwer, Dordrecht, 1997.

\bibitem{Bna}   D. P. Blecher,  {\em A new approach to 
$C^*$-modules,}  Math. Annalen {\bf 307} (1997), 253-290.

\bibitem{Bunpub}  D. P. Blecher, {\em Unpublished notes
on multiplier operator algebras}, (2000).  

\bibitem{BShi}  D. P. Blecher,  {\em  The Shilov boundary
of an operator space, and the characterization
theorems,}  J. Funct. An. {\bf 182} (2001), 280-343.

\bibitem{Bid1}  D. P. Blecher,  {\em 
One-sided ideals and approximate identities in operator algebras,} 
Preprint (September '01).   

 
\bibitem{BEZ}  D. P. Blecher, E. G.
Effros and V. Zarikian, {\em One-sided M-ideals and multipliers 
in operator spaces I,} Preprint (2000).  To appear {\em  Pacific J. Math}.     

\bibitem{BMP} D. P. Blecher, P. S. Muhly and V. I. Paulsen,  
{\em  Categories of
operator modules - Morita equivalence and projective modules,}
 {\em Memoirs of the American Mathematical Society}
Vol. 143, number 681 (Jan, 2000). 

\bibitem{BPnew}  D. P. Blecher and V. I. Paulsen,
 {\em  Multipliers
of operator spaces, and the injective envelope,}
Pacific J. Math.  {\bf 200} (2001), 1-17.
  
%\bibitem{BRS}  D. P.  Blecher, Z. J. Ruan, and A. M.
%Sinclair, {\em A characterization
%of operator algebras}, J. Funct. An. {\bf 89} (1990),
%188-201.

\bibitem{BonsallDuncan}  F. F. Bonsall and J. Duncan,
{\em Complete normed
algebras,}      Springer-Verlag, New York-Heidelberg
(1973).                                                                                         

\bibitem{ERbook}   E. G.
Effros and Z. J. Ruan, {\em Operator Spaces}, Oxford University
Press (2000).                                                             

 
 \bibitem{Ham1}  M. Hamana, {\em    Injective
envelopes of operator systems,}
Publ. R.I.M.S. Kyoto Univ. {\bf 15} (1979), 773-785.
 
\bibitem{Ham2}  M. Hamana, {\em Triple  envelopes and Silov
boundaries of operator spaces,}  Math. J. Toyama University
{\bf 22} (1999), 77-93.   


\bibitem{KP}  M. Kaneda and  V. I. Paulsen, {\em 
Characterizations of essential ideals as operator modules 
over $C^*$-algebras}, Preprint (2000).  
%To appear.

\bibitem{Lin}  H. Lin, {\em Bounded module maps and pure completely
positive maps,}  J. Operator Theory {\bf 26} (1991), 121-138. 

%\bibitem{eor}   E. G.
%Effros, N. Ozawa and Z. J. Ruan, {\em On injectivity and
%nuclearity for operator spaces,} Duke Math. J. To appear.
 
\bibitem{Pal}  T. W. Palmer, {\em Banach Algebras and the general 
theory of *-algebras Vol. I,} Encyclopedia of mathematics and its
applications Vol. 49, Cambridge University Press (1994).

\bibitem{P}   V. I. Paulsen, {\em Completely bounded maps 
and dilations,} Pitman Research
Notes in Math., Longman, London, 1986.

\bibitem{Punp}  V. I. Paulsen, {\em Unpublished notes on 
multiplier algebras,} (circa 1992).

\bibitem{Ped} G. Pederson, {\em C$^*$-algebras
and their automorphism groups,}
Academic Press (1979).                         
 
\bibitem{Pis} G. Pisier, {\em Introduction to the theory
of operator spaces,} to appear.

\bibitem{PuR}  Y-t. Poon and Z. J. Ruan, {\em Operator algebras with
contractive approximate identities, } Canadian J. Math. {\bf 46}(1994),
397-414.     

%\bibitem{W-O}  N. E. Wegge-Olsen, {\em K-theory and $C^*$-algebras,}
% Oxford Univ. Press (1993).
                          
                         
\end{thebibliography}
\end{document} 

math0111122.extracted_bib
\begin{thebibliography}{99}
 
\bibitem{Arv1}  W. B. Arveson,
{\em Subalgebras of }$C^{*}-${\em algebras,}
 Acta Math. {\bf 123 }(1969), 141-224.

 
%\bibitem{Bhmo} D. P. Blecher,  
%{\em  A generalization of Hilbert modules,}
%J. Funct. An. {\bf 136} (1996), 365-421.    
\bibitem{Bnat}  D. P. Blecher,  {\em Some general theory of
operator algebras and their modules,} in
 {\em Operator algebras and
applications\/},
A. Katavalos (editor), NATO ASIC, Vol.
495, Kluwer, Dordrecht, 1997.

\bibitem{Bna}   D. P. Blecher,  {\em A new approach to 
$C^*$-modules,}  Math. Annalen {\bf 307} (1997), 253-290.

\bibitem{Bunpub}  D. P. Blecher, {\em Unpublished notes
on multiplier operator algebras}, (2000).  

\bibitem{BShi}  D. P. Blecher,  {\em  The Shilov boundary
of an operator space, and the characterization
theorems,}  J. Funct. An. {\bf 182} (2001), 280-343.

\bibitem{Bid1}  D. P. Blecher,  {\em 
One-sided ideals and approximate identities in operator algebras,} 
Preprint (September '01).   

 
\bibitem{BEZ}  D. P. Blecher, E. G.
Effros and V. Zarikian, {\em One-sided M-ideals and multipliers 
in operator spaces I,} Preprint (2000).  To appear {\em  Pacific J. Math}.     

\bibitem{BMP} D. P. Blecher, P. S. Muhly and V. I. Paulsen,  
{\em  Categories of
operator modules - Morita equivalence and projective modules,}
 {\em Memoirs of the American Mathematical Society}
Vol. 143, number 681 (Jan, 2000). 

\bibitem{BPnew}  D. P. Blecher and V. I. Paulsen,
 {\em  Multipliers
of operator spaces, and the injective envelope,}
Pacific J. Math.  {\bf 200} (2001), 1-17.
  
%\bibitem{BRS}  D. P.  Blecher, Z. J. Ruan, and A. M.
%Sinclair, {\em A characterization
%of operator algebras}, J. Funct. An. {\bf 89} (1990),
%188-201.

\bibitem{BonsallDuncan}  F. F. Bonsall and J. Duncan,
{\em Complete normed
algebras,}      Springer-Verlag, New York-Heidelberg
(1973).                                                                                         

\bibitem{ERbook}   E. G.
Effros and Z. J. Ruan, {\em Operator Spaces}, Oxford University
Press (2000).                                                             

 
 \bibitem{Ham1}  M. Hamana, {\em    Injective
envelopes of operator systems,}
Publ. R.I.M.S. Kyoto Univ. {\bf 15} (1979), 773-785.
 
\bibitem{Ham2}  M. Hamana, {\em Triple  envelopes and Silov
boundaries of operator spaces,}  Math. J. Toyama University
{\bf 22} (1999), 77-93.   


\bibitem{KP}  M. Kaneda and  V. I. Paulsen, {\em 
Characterizations of essential ideals as operator modules 
over $C^*$-algebras}, Preprint (2000).  
%To appear.

\bibitem{Lin}  H. Lin, {\em Bounded module maps and pure completely
positive maps,}  J. Operator Theory {\bf 26} (1991), 121-138. 

%\bibitem{eor}   E. G.
%Effros, N. Ozawa and Z. J. Ruan, {\em On injectivity and
%nuclearity for operator spaces,} Duke Math. J. To appear.
 
\bibitem{Pal}  T. W. Palmer, {\em Banach Algebras and the general 
theory of *-algebras Vol. I,} Encyclopedia of mathematics and its
applications Vol. 49, Cambridge University Press (1994).

\bibitem{P}   V. I. Paulsen, {\em Completely bounded maps 
and dilations,} Pitman Research
Notes in Math., Longman, London, 1986.

\bibitem{Punp}  V. I. Paulsen, {\em Unpublished notes on 
multiplier algebras,} (circa 1992).

\bibitem{Ped} G. Pederson, {\em C$^*$-algebras
and their automorphism groups,}
Academic Press (1979).                         
 
\bibitem{Pis} G. Pisier, {\em Introduction to the theory
of operator spaces,} to appear.

\bibitem{PuR}  Y-t. Poon and Z. J. Ruan, {\em Operator algebras with
contractive approximate identities, } Canadian J. Math. {\bf 46}(1994),
397-414.     

%\bibitem{W-O}  N. E. Wegge-Olsen, {\em K-theory and $C^*$-algebras,}
% Oxford Univ. Press (1993).
                          
                         
\end{thebibliography}