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\sloppy
\date{}
\author{V.~M.~Manuilov and K. Thomsen}
\title[Asymptotic homomorphisms and $C^*$-extensions]%
{Asymptotic homomorphisms and $C^*$-extensions%
%\footnote{Partially supported by RFBR (grant No 99-01-01201).}
}
\begin{document}
\maketitle
%\footnotetext[1]{Partially supported by RFBR, grant No 99-01-01201}
% \begin{abstract}
% \end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Introduction}
In this note we provide a very brief and strongly biased introduction to
two theories involving $C^*$-algebras, --- that of asymptotic homomorphisms
and that of extensions. The aim is to describe and give perspective to our
work on the relation between the two theories, and to formulate two
questions which arise from this work.
The first section is devoted to asymptotic homomorphisms,
the second to extensions and the third to some of the constructions
relating them.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Asymptotic homomorphisms of $C^*$-algebras}\label{asym}
In \cite{Connes-Higson} Connes and Higson introduced the following notion.
\begin{dfn}[\cite{Connes-Higson}]\label{dfn_asymptotic}
{\rm
Let $A$ and $B$ be $C^*$-algebras. An {\it asymptotic homomorphism} from
$A$ to $B$ is a family of maps
$(\varphi_t)_{t\in[0,\infty)}:A\to B$ with the following properties:
\begin{enumerate}
\item
for any $a\in A$ the map $[0,\infty)\to B$ given by $t\mapsto\varphi_t(a)$
is continuous;
\item
for any $a,b\in A$, $\lambda\in{\Bbb C}$ one has
\begin{itemize}
\item
$\lim_{t\to\infty}\|\varphi_t(a^*)-\varphi_t(a)^*\|=0$;
\item
$\lim_{t\to\infty}\|\varphi_t(a+\lambda
b)-\varphi_t(a)-\lambda\varphi_t(b)\|=0$;
\item
$\lim_{t\to\infty}\|\varphi_t(ab)-\varphi_t(a)\varphi_t(b)\|=0$.
\end{itemize}
\end{enumerate}
}
\end{dfn}
Note that the individual members, the $\varphi_t$'s, in an asymptotic
homomorphism need not preserve any structure; it's only their
asymptotic behaviour that matters.
\begin{ex}\label{ex}
\rm{To give an example of an asymptotic homomorphism, consider the so-called
Voiculescu matrices, \cite{Voiculescu}: For any natural number $n>2$, put
$\lambda_n={\rm e}^{2\pi i/n}$ and let $u_n,v_n\in {\rm U}_n$ be
two unitaries,
{\small
$$
u_n=
\left(\begin{array}{cccccc}
\lambda_n&&&&&0\\
&\lambda_n^2&&&&\\
&&\cdot&&&\\
&&&\cdot&&\\
&&&&\cdot&\\
0&&&&&\lambda_n^n
\end{array}\right),
\qquad
v_n=
\left(\begin{array}{cccccc}
0&0&\cdot&\cdot&0&1\\
1&0&0&\cdot&\cdot&0\\
0&1&0&0&\cdot&\cdot\\
&&\cdot&\cdot&\cdot&\\
0&\cdot&\cdot&1&0&0\\
0&0&\cdot&\cdot&1&0
\end{array}\right).
$$ }
Denote by $\|\cdot\|$ the standard operator norm on the space $M_n$ of
complex $n{\times}n$ matrices, and note that
$\|u_nv_n-v_nu_n\|=|\lambda_n-1|$ tends to zero as $n\to\infty$. Moreover,
there exists a path $(u_t,v_t)$,
connecting the pair $(u_n\oplus 1,v_n\oplus 1)$ with the pair
$(u_{n+1},v_{n+1})$ in ${\rm U}_{n+1}$, such that the norm of the
commutator $\|[u_t,v_t]\|$ does not exceed $|\lambda_n-1|$ along the whole
path, \cite{Mish-Noor}. Thus there are two continuous paths of
unitaries, $u_t$ and $v_t$,
parametrized by $t \in [0,\infty)$ in the $C^*$-algebra
$\K^+$ of compact operators with adjoined unit such that
\begin{equation*}\label{A}
\lim_{t \to \infty} u_tv_t - v_tu_t = 0,
\end{equation*}
and such that $(u_n,v_n)$ are the Voiculescu matrices for all $n$. (We are
here
considering the unitary group of $M_n$ as a subgroup of the unitary group
$\K^+$
in the natural way.) To obtain an asymptotic homomorphism from these paths,
consider
for any $C^*$-algebra $B$, the $C^*$-algebra $C_b([0,\infty);B)$ of bounded
continuous
$B$-valued functions on $[0,\infty)$ and let $C_0([0,\infty);B)$ be the
ideal of
functions vanishing at infinity. There are then unitaries $U,V
\in C_b([0,\infty);\K^+)$ given by $U(t) = u_t$ and $V(t) = v_t$. By
(\ref{A}) the image of $U$ and $V$ in the quotient
$C_b([0,\infty);B)/C_0([0,\infty);B)$ are commuting, and define
therefore a $*$-homomorphism
$\varphi : C(\mathbb T^2) \to C_b([0,\infty);B)/C_0([0,\infty);B)$. By
Michael's
selection theorem, \cite{Michael, Bartle-Graves, Loring}, there is a
continuous right inverse $s$
for the quotient map $C_b([0,\infty);B) \to
C_b([0,\infty);B)/C_0([0,\infty);B)$. We
can then define a family, $\bold\Phi_t, t \in [0,\infty)$, of
maps $C(\mathbb T^2) \to \K^+$ by
$$
\bold\Phi_t(f) = s(\varphi(f))(t).
$$
Since $\varphi$ is a $*$-homomorphism and $s$ is continuous,
$(\varphi_t)_{t \in [0,\infty)}$, is an asymptotic homomorphism. By
construction
$$
\lim_{n \to \infty} \|\bold\Phi_n(z_1) - u_n\| = \lim_{n \to \infty}
\|\bold\Phi_n(z_2) - v_n\| = 0,
$$
where $z_1$ and $z_2$ are the unitaries in $C(\mathbb T^2)$ given
by the projections $\mathbb T^2 \to \mathbb T$ to the first and second
coordinate, respectively. This asymptotic homomorphism is highly non-trivial;
in particular, it is not asymptotic to a genuine $*$-homomorphism
$C(\mathbb T^2) \to \K^+$. This can be seen by using the proof of Voiculescu
from \cite{Voiculescu}, where he shows that the pairs $(u_n,v_n)$ are not
close to any pair of commuting unitaries.}
\end{ex}
Let $B[0,1]$ be the $C^*$-algebra of
continuous $B$-valued functions on $[0,1]$ and let $ev_0,ev_1:B[0,1]\to
B$ be the evaluation homomorphisms at the end-points. Two asymptotic
homomorphisms
$(\varphi^0_t)_{t\in[0,\infty)},(\varphi^1_t)_{t\in[0,\infty)}:A\to B$
are {\it homotopic} when there exists an asymptotic homomorphism
$(\Lambda_t)_{t\in[0,\infty)}:A\to B[0,1]$ such that $ev_i\circ
\Lambda_t=\varphi^i_t$ for $i=0,1$ and all $t$. Denote by $[[A,B]]$ the
set of homotopy classes of asymptotic
homomorphisms from $A$ to $B$. Even up to homotopy, there are in general
many more asymptotic homomorphisms than genuine $*$-homomorphisms; for
example the asymptotic homomorphism $\bold\Phi$
from Example \ref{ex} is not homotopic to a $*$-homomorphism.
Assume now that the $C^*$-algebra $B$
is stable, i.e. satisfies that $B \otimes \mathcal K \simeq B$. Then
$[[A,B]]$ has an
additive structure making it into an
abelian semi-group. To obtain a group it is in general
necessary to {\it suspend} $A$; the suspension of $A$ is the $C^*$-algebra
$SA = C_0(0,1) \otimes A$, of continuous $A$-valued functions on $[0,1]$
that
vanish at the endpoints. By iteration, we set $S^nA = S(S^{n-1}A)$. The sets
$[[S^nA,B]], n = 1,2,3, \cdots $, are then all groups, in fact they are the
$E$-theory groups of \cite{Connes-Higson}. (For $n = 1$, this fact is due
to Dadarlat and Loring, \cite{Dadarlat}.) It turns out, that due to
Bott-periodicity
in $E$-theory
there are actually only two
groups here, namely $[[SA,B]]$ and $[[S^3A,B]]$. This is because there is
an isomorphism
\begin{equation}\label{bott}
Bott : [[S^3A,B]]\cong [[SA,B]]
\end{equation}
which can be obtained, for example, from the asymptotic homomorphism
$\bold\Phi$ of Example \ref{ex}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extensions of $C^*$-algebras}
A sequence
\begin{equation}\label{short_exact}
\begin{xymatrix}{
0\ar[r] & B\ar[r] & E\ar[r] & A\ar[r] & 0
}
\end{xymatrix}
\end{equation}
of $C^*$-algebras and $*$-homomorphisms is called a {\it short exact}
sequence if the kernel of any $*$-homomorphism in (\ref{short_exact})
coincides with the range of the previous $*$-homomorphism. This simply
means that $B$ is an ideal in $E$ and the corresponding quotient is $A$.
Given $C^*$-algebras $A$ and
$B$, a short exact sequence (\ref{short_exact}) is called an {\it
extension} of $A$ by $B$. The extensions of $A$ by $B$ are thus the
possible ways one can build a $C^*$-algebra with ideal $B$ and quotient
$A$, and because of the analogy with similar objects from other categories
(e.g. groups), it should not be a surprise that the structure of such
extensions is rich and complicated. The first general study of the
extensions of two given $C^*$-algebras was performed by Busby \cite{Busby},
who introduced the fundamental tool for this purpose, the so-called Busby
invariant. To explain, what it is, let us
recall the notion of the multiplier algebra. The simplest approach uses
the notion of Hilbert $C^*$-modules \cite{Paschke,Lance,MaTroJMS}.
Let $B$ be a $C^*$-algebra and let
$(\M,\|\cdot\|_\M)$ be a Banach space whicht is also a (right)
$B$-module equipped with a sesquilinear $B$-valued inner product
$\langle\cdot,\cdot\rangle: \M\times\M\to B$ such that the
following properties hold for any $x,y\in\M$ and $b\in B$:
\begin{enumerate}
\item
$\langle x,yb\rangle=\langle x,y\rangle b$;
\item
$\langle x,y\rangle^*=\langle y,x\rangle$;
\item
$\langle x,x\rangle\in B$ is a positive element;
%if $\langle x,x\rangle=0$ then $x=0$;
\item
$\|x\|_\M=\|\langle x,x\rangle\|^{1/2}$.% for any $x\in\M$.
\end{enumerate}
Two basic examples of Hilbert $C^*$-modules are $B$ itself (with the inner
product $\langle a,b\rangle=a^*b$ for $a,b\in B$) and the standard module
$l_2(B)=\oplus_{n=1}^\infty B=B^{\mathbb N}$ consisting of sequences
$(b_n)_{n\in{\Bbb N}}$ of elements $b_n\in B$ such that the series
$\sum_n b_n^*b_n$ is norm convergent in $B$. The inner product is then
given by $\langle
(a_n),(b_n)\rangle=\sum_n a_n^*b_n$ for $(a_n),(b_n)\in\M$.
A linear operator $T:\M\to\M$ is called {\it adjointable} when there a
linear operator
$T^*:\M\to\M$ such that $\langle T(m),n\rangle=\langle
m,T^*(n)\rangle$ for any $m,n\in\M$. The set of adjointable operators
$\mathbb L(\M)$ has a natural structure of a $C^*$-algebra; the norm is
the operator norm and the
adjoint is given by $T \mapsto T^*$.
An operator $T\in\mathbb L(\M)$
is called an elementary operator if it has the form $Tx=z\langle
y,x\rangle$ for some $y,z\in\M$. The closed linear span $\mathbb K(\M)$
of elementary operators is the $C^*$-algebra of `compact' operators
on $\M$. There are natural identifications $\mathbb K(B)\cong B$ and
$\mathbb L(l_2(B))\cong B\otimes\K$, where $\mathcal K$
is the $C^*$-algebra of compact operators on the Hilbert space $l_2$.
For a $C^*$-algebra $B$ the multiplier $C^*$-algebra $M(B)$ can be defined
as $\mathbb L(B)$. It is easy to check that $M(B)$
contains $B=\mathbb K(B)$ as an ideal. The corresponding quotient
$C^*$-algebra
is called a {\it corona} (or a generalized Calkin) algebra and is denoted
by $Q(B)=M(B)/B$. Recall that an ideal $B$ in $E$ is essential if the
conditions $e\cdot B=0$ and $B\cdot e=0$ imply $e=0$ for any $e\in E$.
The multiplier $C^*$-algebra is known to be the biggest $C^*$-algebra that
contains $B$ as an essential ideal, \cite{Pedersen}.
If $B$ is an ideal in a $C^*$-algebra $E$, then the left multiplication by
an element $e\in E$ defines an adjointable $B$-linear operator on $B$ and
thus one obtains a $*$-homomorphism $\mu:E\to M(B)$. Since $\mu(B)\subset B$,
$\mu$ generates a $*$-homomorphism $\tau:A=E/B\to Q(B)$. This
$*$-homomorphism is called the {\it Busby invariant} for the extension
(\ref{short_exact}). One can retrieve the extension
from its Busby invariant by the following procedure. Let $q:M(B)\to Q(B)$
denote
the quotient map.
For any $*$-homomorphism $\tau:A\to Q(B)$ the $C^*$-algebra
$E=\{(a,m)\in A\times M(B): q(m)=\tau(a)\}$ is an extension of $A$ by $B$
whose
Busby invariant is $\tau$. The Busby invariant
allows us to identify extensions of $A$ by $B$ with $*$-homomorphisms from
$A$ to $Q(B)$.
The breakthrough in the study of extensions of $C^*$-algebras came with the
work
of Brown, Douglas and Fillmore, who developed the theory to handle a
specific problem in operator theory, and only later realized the
connection to K-homology, cf. \cite{BDF0},\cite{BDF}. Their work is a
milestone and a brilliant unification of analytical and topological
methods. Brown, Douglas and Fillmore introduced a crucial equivalence
relation
on extensions which gives the corresponding equivalence classes the
structure of
an abelian semigroup, at least when the algebra $B$ which is the ideal in
the
extensions, is stable, i.e. satisfies that $B \otimes \mathcal K \simeq B$.
In
this case there is a natural embedding $Q(B) \oplus Q(B) \subseteq Q(B)$
which
allows one to define the direct sum of two extensions. In more detail the
addition comes around as follows: Since $B$ is stable there is
pair $V_1,V_2$ of isometries in $M(B)$ such that $V_1^*V_2 = 0$ and
$V_1V_1^* + V_2V_2^*$ and we can define the addition $\varphi + \psi$ of
two extensions $\varphi, \psi : A \to Q(B)$ as the map
$$
a \mapsto q(V_1)\varphi(a)q(V_1)^* + q(V_2)\psi(a)q(V_2)^* .
$$
This will make the extensions of $A$ and $B$ into an abelian semi-group,
provided
we mod out by an appropriate equivalence relation. The strongest relation
that
serves this purpose is {\it unitary equivalence}; two
extensions $\varphi, \psi : A \to Q(B)$ are {\it unitarily equivalent} when
there
is a unitary $U \in M(B)$ such that $\Ad q(U) \circ \varphi = \psi$. The
unitary
equivalence classes of extensions of $A$ by $B$ is then an abelian
semi-group and
the addition is independent of the choice of isometries. Note, however, that
there is in general no neutral element in this semi-group.
While we are forced to identify unitarily equivalent extensions if we want to
have the semi-group structure, there are many choices one can make to
get a group structure. One can introduce weaker equivalence classes
that respect the addition --- and there is a great variety of possibilities
here --- or
one can restrict the attention to special classes of extensions. We refer
to the
book of
Blackadar, \cite{Blackadar}, for a brief discussion of some of the
possibilities --- the creative reader can easily extend the list. The most
useful approach has been to consider the quotient by the sub-semigroup
of split extensions, i.e. the one for which the quotient map admits
a $*$-homomorphism as a right inverse, and then restrict attention to the
invertible elements in the resulting quotient semi-group. The result is in
then the KK-theory of Kasparov, \cite{Kasparov},\cite{Arveson}. However,
if we are
not inclined to reduce our ambition and want to study all extensions of
$A$ by $B$, with
the least possible loss of information, we must first consider the
cancellation
semi-group obtained from the semi-group of unitary equivalence
classes of extensions. Indeed, any map into a group must factor through
this
cancellation semi-group which we will denote by
$$
\Ext(A,B)
$$
in the following. By definition two extensions $\varphi, \psi$ of $A$ by
$B$ define
the same element in $\Ext(A,B)$ if and only if there is
a third extension $\lambda$ such that $\varphi + \lambda$ is unitarily
equivalent to
$\psi + \lambda$. Note that $\Ext(A,B)$ has a neutral element, namely
$0$ - the
extension represented by the direct sum of $A$ and $B$. In fact, any
extension
$\lambda$ for which there is another extension $\varphi$ such that
$\lambda + \varphi$
is unitarily equivalent to $\varphi$ must represent the neutral element
$0$ in
$\Ext(A,B)$. In particular, all split extensions, as well as the
asymptotically
split extensions considered in \cite{MT2} and \cite{MT3}, must all represent
the neutral element $0$ in $\Ext(A,B)$.
It follows from \cite{MT3} that when $A$ is a suspension, i.e. is of the
form $A = SD$, there is nothing more to be done; one already
has a group in this case:
\begin{thm}\label{TH1} Let $A$ and $B$ be separable $C^*$-algebras, $B$
stable. Then $\Ext(SA,B)$ is a group.
\end{thm}
\begin{proof}
Consider an extension $\varphi$ of $SA$ by $B$. Let $\alpha $
be the automorphism of
$SA$ obtained by reversing the direction in $(0,1)$; $\alpha(f)(t) =
f(1-t)$.
By Corollary 2.3 of \cite{MT3}, $\varphi + (\varphi \circ \alpha) +0 $ is
asymptotically
split, i.e. there is an asymptotic homomorphism
$\pi = (\pi_t)_{t \in [1,\infty)} : SA \to M(B)$ such that
$q \circ \pi_t = \varphi + (\varphi \circ \alpha) + 0$
for all $t \in [1,\infty)$. Since $B$ is stable there is a sequence
$W_k, k = 1,2, \cdots ,$ of isometries in $M(B)$ such that
$W_k^*W_l = 0,k\neq l$, and such that $\sum_{k=1}^{\infty} W_kW_k^* = 1$,
with
convergence
in the strict topology. Now choose a
discretization $\pi_{t_n},n \in \mathbb N$, of $\pi$, and define an
extension $\lambda : SA \to Q(B)$ by
$$
\lambda(a) = q(\sum_{i=1}^{\infty} W_i\pi_{t_i}(a)W_i^*).
$$
It is then easy to see that there is an isometry $S \in M(B)$ such that
$0 + \lambda = q(S)((q \circ \pi_{t_1}) + \lambda)q(S)^*$. Since
$q \circ \pi_{t_1} = \varphi + (\varphi \circ \alpha) + 0$,
this implies that $\varphi + (\varphi \circ \alpha) + \lambda + 0$ is
unitarily
equivalent
to $ \lambda + 0$, i.e. $\varphi + (\varphi \circ \alpha)$ is $0$ in
the cancellation semi-group.
\end{proof}
This result forces us to raise the following question which we hope
some readers will find just as provoking as the authors:
\bigskip
\emph{Question 1: Is $\Ext(A,B)$ always a group ?}
\bigskip
Any $C^*$-algebra $A$ for which there is a stable $C^*$-algebra $B$ such
that $\Ext(A,B)$ is not a group, can not have the lifting property of
Kirchberg, \cite{Kirchberg}. In particular, as is well-known, it can not
be nuclear.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{From extensions to asymptotic homomorphisms and back}
\label{sectionMichael}
In this section we consider various ways to go from extensions to
asymptotic homomorphisms and back.
Let $B$ be an ideal in a $C^*$-algebra $E$. Recall that an approximate
unit $(u_t)_{t\in[0,\infty)}$ in $B$ is a continuous family of elements
$u_t\in
B$ such that $0 \leq u_t \leq 1$ for all
$t$ and $\lim_{t\to\infty}\|u_tb-b\|=\lim_{t\to\infty}\|bu_t-b\|=0$
for any $b\in B$. An approximate unit $(u_t)_{t\in[0,\infty)}$ in $B$
is called
a {\it quasicentral} (in $E$) when
$\lim_{t\to\infty}\|eu_t-u_te\|=0$ for all $e \in E$. Such a thing
exists always, provided $E$ is separable, \cite{Arveson}. In fact, it
suffices
that the quotient $C^*$-algebra $E/B$ is separable
and that $B$ is $\sigma$-unital (i.e. $B$ contains a countable
approximate unit or, equivalently, a strictly positive element).
To illustrate this notion consider the Toeplitz $C^*$-algebra $T$
in $B(H)$ generated by the unilateral shift (we identify here $H$ with
$l_2({\Bbb N})$). This $C^*$-algebra contains $\K$ as an ideal and the
corresponding quotient $C^*$-algebra is the algebra $C(\mathbb T)$ of
continuous functions on the circle group $\mathbb T$, giving us an
extension of $C(\mathbb T)$ by $\K$: $0 \to \K \to T \to C(\mathbb T) \to 0$.
As an example of a
quasicentral approximate unit one can take for integer values of the
parameter the operators $u_n=\diag\{\underbrace{1,\ldots,1}_{n\ {\rm
times}},\frac{n-1}{n},\frac{n-2}{n},\ldots,\frac{1}{n},0,0,0,\ldots\}$ and
then connect them by a piecewise linear path $u_t$. Obviously $u_t$
asymptotically commutes with the unilateral shift, hence with any element
of $T$.
In order to describe the Connes-Higson construction
\cite{Connes-Higson}, let $\tau:A\to Q(B)$ be an extension. Since
$A$ and $B$ are separable, there is a separable $C^*$-algebra
$E \subseteq M(B)$
such that $B \subseteq E$ and $\tau(A) \subseteq q(E)$. Let $\{u_t\}
\subseteq B$
be an approximate unit for $B$ which is quasi-central for $E$. Let
$f \in C_0(0,1), a \in A$. Pick any $e \in M(B)$ such that $q(e) = \tau(a)$.
We can then define a continuous $B$-valued function $f \otimes a$ on
$[1,\infty )$,
i.e. an element of $C_b([1,\infty), B)$, whose value at $t$ is
$$
f \otimes a(t) = f(u_t)e.
$$
If $e_1 \in M(B)$ is any other element such that $q(e) = \tau(a)$,
$\lim_{t \to \infty} f(u_t)e_1 - f(u_t)e = 0$, so that the image of our
function in $C_b([1,\infty), B)/C_0([1,\infty), B)$ does not depend on which
$e \in M(B) \cap q^{-1}(\tau(a))$ we choose. If we choose $e$ from $E$
we have that
$\lim_{t \to \infty} u_te - eu_t = 0$ since $\{u_t\}$ is quasicentral
in $E$. It follows then easily that the recipe
$$
(f,a) \to f \otimes a + C_0([1,\infty),B)
$$
gives us a $*$-homomorphism
$$
\tilde{\tau} : C_0(0,1) \otimes A \to C_b([1,\infty), B)/C_0([1,\infty), B).
$$
We can now define from $\tau$ an asymptotic homomorphism $CH(\tau) :
SA \to B$ in the following way: For each $x \in SA$, choose a lift
$\hat{\tau}(x) \in C_b([1,\infty), B)$ of $\tilde{\tau}(x)$, and set
$$
CH(\tau)_t(x) = \hat{\tau}(x)(t).
$$
If we assume that $B$ is stable we can consider this construction of
Connes and Higson as a semi-group homomorphism
$$
\begin{xymatrix}{%
CH:\Ext(A,B)\ar[r]&[[SA,B]].
}
\end{xymatrix}
$$
As was pointed out in Theorem \ref{TH1}, it follows from the opening results
of \cite{MT3} that $\Ext(SA,B)$ is always a group. The main result
of \cite{MT3} is the following:
\begin{thm}\label{TH2} $CH : \Ext(SA,B) \to [[S^2A,B]]$ is an isomorphism.
\end{thm}
One consequence of Theorem \ref{TH2} is that it implies that most of the
alternative equivalence relations one can consider on the set of extensions
of $SA$ by $B$ will all give rise to same group. In particular, this is the
case of the homotopy classes: We say that to extensions of $A$ by $B$ are
\emph{homotopic} when they can be realized as the top and buttom
extensions in a commuting diagram of the form
\begin{equation*}
\begin{xymatrix}{
0 \ar[r] & B \ar[r] & E_0 \ar[r] & A \ar[r] & 0 \\
0 \ar[r] & B[0,1] \ar[r] \ar[u]^{ev_0} \ar[d]_{ev_1} & E \ar[r]
\ar[u] \ar[d] & A \ar[r] \ar[u] \ar[d] & 0 \\
0 \ar[r] & B \ar[r] & E_1 \ar[r] & A \ar[r] & 0,
}
\end{xymatrix}
\end{equation*}
where the middle row is an extension of $A$ by $B[0,1]$, and
$ev_i : B[0,1] \to B, i = 0,1$, denote evaluation at $i$. Modulo
homotopy the extensions of $SA$ by $B$ is a group, and by Theorem \ref{TH2}
this
group is isomorphic to $[[S^2A,B]]$ under the Connes-Higson map. It follows
from
the work of Dadarlat and Loring, \cite{Dadarlat}, that $[[S^2A,B]]$ is the
$E$-theory
group of Connes and Higson. Consequently the $E$-theory of Connes and
Higson is
exactly the theory of homotopy classes of extensions, a fact which was
already
anticipated by Connes and Higson in \cite{Connes-Higson}.
In view of Theorem \ref{TH2} it is natural to extend Question 1 above with
the following.
\bigskip
\emph{Question 2: Is $CH : \Ext(A,B) \to [[SA,B]]$ always an isomorphism?}
\bigskip
Of course, Question 2 can only have a positive answer if Question 1 does.
Should
the answer to Question 1 be negative, Question 2 must be rephrased to ask
for a
description of the range and 'kernel' of $CH : \Ext(A,B) \to [[SA,B]]$.
Let us describe the inverse of $CH$ when the quotient map is a suspension.
The
construction of this inverse was inspired by \cite{Mish-Noor,MM}. The first
step is the
following lemma.
\begin{lem}[\cite{MT1}]\label{discretiz}
Let $A$ be a separable $C^*$-algebra and let
$(\varphi_t)_{t\in[0,\infty)}:A\to B$ be an equicontinuous asymptotic
homomorphism.
Then there exists a sequence $(t_n)_{n\in{\Bbb N}}$ in $[0,\infty)$ such
that
$\lim_{n\to\infty}t_n=\infty$ and
$\lim_{n\to\infty}\sup_{t\in[t_n,t_{n+1}]}
\|\varphi_t(a)-\varphi_{t_n}(a)\|=0$
for any $a\in A$.
\end{lem}
Equicontinuity of an asymptotic homomorphism $(\varphi_t)_{t \in [0,\infty)}$
means that
the family of maps $a \mapsto \varphi_t(a)$ is an equicontinuous
family. Every asymptotic homomorphism is
asymptotically equal to one which is equicontinuous, so this
additional requirement in Lemma \ref{discretiz} is harmless. The discrete
asymptotic
homomorphism $(\phi_n)_{n\in{\Bbb N}}$
with $\phi_n=\varphi_{t_n}$, where the sequence $(t_n)_{n\in{\Bbb N}}$ is
as in Lemma \ref{discretiz}, we call a {\it discretization} of the
asymptotic homomorphism $(\varphi_t)_{t\in[0,\infty)}$. Given such a
discretization $(\phi_n)_{n\in{\mathbb N}}$ of
$(\varphi_t)_{t\in[0,\infty)}$, we proceed as
follows. Set $\phi_n(a)=0$ when $n = 0,-1,-2, \cdots $. Consider
the Hilbert $C^*$-module $B^{\mathbb Z}$ consisting of two-sided
sequences $(b_i)_{i \in \mathbb Z}$ in $B$ such that
$\lim_{N \to \infty} \sum_{i = -N}^N b_i^*b_i$ converges in $B$. We will use
the standard matrix
notation; the elementary matrices $e_{i,j},i,j \in \mathbb Z$, are the
operators
$e_{i,j}: B^{\mathbb Z} \to B^{\mathbb Z}$ in $\mathbb L(B^{\mathbb Z})$
given by
$$
e_{i,j}((b_k))_l = \begin{cases} b_j, & \ l = i, \\ 0, & \ l \neq i.
\end{cases}
$$
We can then define a map $\phi : A \to \mathbb L(B^{\mathbb Z})$ by
$$
\phi(a) = \sum_{i \in \mathbb Z} \phi_n(a)e_{i,i},
$$
and a unitary $T \in \mathbb L(B^{\mathbb Z})$ by
$$
T=\sum_{n=-\infty}^\infty e_{n,n-1}.
$$
$T$ is bilateral shift of $B$-coordinates. Since $\phi(a)$ and $T$ commutes
modulo $\mathbb K(B^{\mathbb Z})$, we can define a $*$-homomorphism
$E(\varphi)
: C(\mathbb T) \otimes A \to \mathbb L(B^{\mathbb Z})/\mathbb
K(B^{\mathbb Z})$
such that
$$
E(\varphi) (f \otimes a) = f(T)\phi(a) + \mathbb K(B^{\mathbb Z}) .
$$
By using that $B^{\mathbb Z} \simeq B$ as Hilbert $B$-algebras because $B$ is
stable, we can consider $E(\varphi)$ as a $*$-homomophism
$C(\mathbb T)\otimes A \to Q(B)$, i.e. as an extension of
$C(\mathbb T)\otimes A$
by $B$. Since $SA \simeq \{f \in C(\mathbb T) \otimes A : f(1) = 0\}$, we
obtain
an extension of $SA$ by $B$ by restriction. It turns that the class of this
restriction $E(\varphi)|_{SA}$ in $\Ext(SA,B)$ does not depend on the
choice of
discretization and in fact depends only on the homotopy class of $\varphi$.
In this
way we get a semi-group homomorphism
$$
E:[[A,B]]\to \Ext(SA,B).
$$
Again the situation is much improved when $A$ is suspended. The following
result
was obtained in \cite{MT3}.
\begin{thm}\label{TH3} $E : [[SA,B]] \to \Ext(S^2A,B)$ is an isomorphism,
and the diagram
\begin{equation*}
\begin{xymatrix}{
& \Ext(S^2A,B) \ar[dr]^-{CH} & \\
[[SA,B]] \ar[ur]^-E \ar[rr]^-{Bott} & & [[S^3A,B]] }
\end{xymatrix}
\end{equation*}
commutes
\end{thm}
Thus $E$ is the inverse of $CH$, up to Bott periodicity.
%{\small%\footnotesize
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%}
\vspace{2cm}
\parbox{7cm}{V. M. Manuilov\\
Dept. of Mech. and Math.,\\
Moscow State University,\\
Moscow, 119899, Russia\\
e-mail: manuilov@mech.math.msu.su
}
\hfill
\parbox{6cm}{K. Thomsen\\
Institut for matematiske fag,\\
Ny Munkegade, 8000 Aarhus C,\\
Denmark\\
e-mail: matkt@imf.au.dk
}
\end{document}