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\def\C{{$C^*$-algebra}}
\sloppy
\date{}
%\date{10 August 1997}
\author{V.~M.~Manuilov}
\title{Asymptotic homomorphisms into the Calkin algebras%
\footnote{This research was partially supported by
RFBR (grant No 99-01-01201).}}
\begin{document}
\maketitle
\begin{abstract}
Let $A$ be a separable $C^*$-algebra and let $B$ be a $C^*$-algebra
with a strictly positive element. We consider the (semi)group
$\Ext^{as}(A,B)$ (resp. $\Ext(A,B)$) of homotopy classes of asymptotic
(resp. of genuine) homomorphisms from $A$ to the Calkin algebra
$Q(B\otimes\K)=\M(B\otimes\K)/B\otimes\K$
and the natural map $i:\Ext(A,B)\ar\Ext^{as}(A,B)$. We show that if $A$ is
a suspension then $\Ext^{as}(A,B)$ coincides with $E$-theory of Connes and
Higson and the map $i$ is an isomorphism. In particular any
homotopy class of asymptotic homomorphisms from $SA$ to $Q(B\otimes\K)$
contains some genuine homomorphism.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\protect\small\protect
\section{Introduction}
}
The old problem of describing extensions of $C^*$-algebras was a stimulus
for penetration of topological methods into $C^*$-algebra theory
\cite{BDF}. In \cite{Kasparov} it was shown that for nuclear
$C^*$-algebras their extensions can be described in terms of the
Kasparov's $KK$-functor. Unfortunately there are still no results beyond
the nuclear case. We introduce a group of so-called phantom extensions
(possibly empty) and show that when $A$ is suspended then the group of
extensions $\Ext(A,B)$ naturally decomposes into a direct sum of
the $E$-group \cite{Connes-Higson} of asymptotic homomorphisms and the
group of phantom extensions.
Let $A$, $B$ be $C^*$-algebras. Remind \cite{Connes-Higson} that a
collection of maps
$$
\varphi=(\varphi_t)_{t\in[1,\i)}:A\ar B
$$
is called an asymptotic homomorphism if for every $a\in A$ the map
$t\mapsto\varphi_t(a)$ is continuous and
if for any $a,b\in A$, $\lambda\in{\bf C}$ one has
$$
\lim_{t\to\i}\norm{\varphi_t(ab)-\varphi_t(a)\varphi_t(b)}=0;
$$
$$
\lim_{t\to\i}\norm{\varphi_t(a+\lambda b)-\varphi_t(a)-\lambda\varphi_t(b)}=0;
$$
$$
\lim_{t\to\i}\norm{\varphi_t(a^*)-\varphi_t(a)^*}=0.
$$
Two asymptotic homomorphisms $\varphi^{(0)}$ and $\varphi^{(1)}$ are
homotopic if there exists an asymptotic homomorphism $\Phi$ from $A$ to
$B\otimes C[0,1]$ such that its compositions with the evaluation maps at
$0$ and at $1$ coincide with $\varphi^{(0)}$ and $\varphi^{(1)}$
respectively. The set of homotopy classes of asymptotic homomorphisms
from $A$ to $B$ is denoted by $[[A,B]]$
\cite{Connes-Higson,Dadarlat-Loring}.
Throughout this paper we always assume that $A$
is separable and that $B$ has a strictly positive element.
By $\Ext(A,B)$ we denote the set of homotopy classes of extensions of $A$ by
$B\otimes\K$, where $\K$ denotes the $C^*$-algebra of compacts.
We identify extensions with homomorphisms into the Calkin
algebra $Q(B\otimes\K)=\M(B\otimes\K)/B\otimes\K$
by the Busby invariant \cite{Busby}. Two extensions
$f_0,f_1:A\ar Q(B\otimes\K)$ are homotopic if there exists an extension
$F:A\ar Q(B\otimes\K\otimes C[0,1])$
such that its composition with the evaluation
maps at $0$ and at $1$ coincide with $f_0$ and $f_1$ respectively.
Similarly we denote by $\Ext^{as}(A,B)$ the set of homotopy classes of
asymptotic homomorphisms from $A$ to $Q(B\otimes\K)$.
Two asymptotic homomorphisms
$$
\varphi^{(i)}=(\varphi_t^{(i)})_{t\in[1,\i)}:A\ar Q(B\otimes\K),
\quad i=0,1,
$$
are homotopic if there exists an asymptotic homomorphism
$\Phi=(\Phi_t)_{t\in[1,\i)}:A\ar Q(B\otimes\K\otimes C[0,1])$ such that its
compositions with the evaluation maps at $0$ and at $1$ coincide with
$\varphi^{(0)}$ and $\varphi^{(1)}$ respectively. Asymptotic homomorphisms
into $Q(B\otimes\K)$ we call sometimes {\it asymptotic extensions}.
All these sets are equipped with a natural group structure when $A$ is a
suspension, i.e. $A=SD=C_0({\bf R})\otimes D$ for some $C^*$-algebra $D$.
We usually identify the $C^*$-algebras $C_0({\bf R})$ and $C_0(0,1)$.
As every genuine homomorphism can be viewed as an asymptotic one,
so we have a natural map
\be\label{i}
i:\Ext(A,B)\ar\Ext^{as}(A,B).
\ee
It is well-known that usually there is much more asymptotic homomorphisms
than genuine ones, e.g. for $A=C_0({\bf R}^2)$ all
genuine homomorphisms of $A$ into $\K$ are homotopy trivial though
the group $[[C_0({\bf R}^2),\K]]$ coincides with $K_0(C_0({\bf
R}^2))={\bf Z}$ via the Bott isomorphism.
On the other hand there are cases known when all asymptotic homomorphisms
are asymptotically unitarily equivalent to genuine ones \cite{Phillips}.
The main purpose of the paper is to prove surjectivity of the map (\ref{i})
when $A$ is a suspension. This makes a contrast with the case of
mappings into the compacts. As a by-product we get
another description of the $E$-theory in terms of asymptotic extensions.
The main tool in this paper is the Connes--Higson map \cite{Connes-Higson}
$$
CH:\Ext(A,B)\ar [[SA,B\otimes\K]],
$$
which plays an important role in $E$-theory.
Remind that for $f\in\Ext(A,B)$
this map is defined by
$CH(f)=(\varphi_t)_{t\in[1,\i)}$, where $\varphi$ is given by
$$
\varphi_t:\alpha\otimes a\longmapsto \alpha(u_t)f'(a), \qquad
a\in A, \ \alpha\in C_0(0,1).
$$
Here $f':A\ar \M(B\otimes\K)$ is a set-theoretic lifting for
$f:A\ar Q(B\otimes\K)$ and $u_t\in B\otimes\K$
is a quasicentral approximate unit
\cite{Arveson} for $f'(A)$. We are going to show that by fine tuning of
this quasicentral approximate unit one can define also a map
$$
\til{CH}:\Ext^{as}(A,B)\ar [[SA,B\otimes\K]]
$$
extending $CH$ and completing the commutative triangle diagram
$$
\diagram
\Ext(A,B)\rto^i\drto_{CH}&\Ext^{as}(A,B)\dto^{\til{CH}}\\
&[[SA,B\otimes\K]].&
\enddiagram
%
%\begin{array}{ccc}
%\Ext(A,B)\!\!\!\!\!&\stackrel{i}{\ar}&\!\!\!\!\!\Ext^{as}(A,B)\\
%&\!\!\!\!\!\!\!\!\!\!{\scriptstyle CH}\searrow&
%\downarrow{\scriptstyle \til{CH}}\\
%&&\!\!\!\!\![[SA,B\otimes\K]].
%\end{array}
$$
We will show that the map $\til{CH}$ is an isomorphism when $A$ is a
suspension.
I am grateful to K.~Thomsen for his hospitality during my visit
to \AA rhus University in 1999 when the present paper was conceived.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\protect\small\protect
\section{An extension of the Connes--Higson map}
}
A useful tool for working with asymptotic homomorphisms is the
possibility of discretization suggested in
\cite{Mish-Noor,Man-Mish,Man-Th1}.
Let $\Ext^{as}_{discr}(A,B)$ denote the set of homotopy classes of
discrete asymptotic homomorphisms $\varphi=(\varphi_n)_{n\in{\bf N}}:A\ar
Q(B\otimes\K)$
with the additional crucial property suggested by Mishchenko: for
every $a\in A$ one should have
\be\label{n+1}
\lim_{n\to\i}\norm{\varphi_{n+1}(a)-\varphi_n(a)}=0.
\ee
In a similar way we define a set $[[A,B]]_{discr}$ as a set of homotopy
classes of discrete asymptotic homomorphisms with the property
(\ref{n+1}). Without loss of generality we can assume that the sequence
$(\varphi_n)_{n\in{\bf N}}$ is equicontinuous \cite{Loring}.
\begin{lem}\label{cont=discr}
One has
$[[A,B]]=[[A,B]]_{discr}$, \ $\Ext^{as}(A,B)=\Ext^{as}_{discr}(A,B)$.
\end{lem}
\proof
The first equality is proved in \cite{Man-Th1}. The second one can be
proved in the same way. For an asymptotic
homomorphism $\varphi=(\varphi_t)_{t\in[1,\i)}:A\ar Q(B\otimes\K)$
one can find an
infinite sequence of points $\{t_i\}_{i\in{\bf N}}\subset[1,\i)$
satisfying the following properties
\begin{enumerate}
\item
the sequence $\{t_i\}_{i\in{\bf N}}$ is non-decreasing and approaches
infinity;
\item
for every $a\in A$ one has \quad
${\displaystyle
\lim_{i\to\i}\sup_{t\in[t_i,t_{i+1}]}\norm{\varphi_t(a)-\varphi_{t_i}(a)}=0.
}
$
\end{enumerate}
Then $\phi=(\varphi_{t_i})_{i\in{\bf N}}$ is a discrete asymptotic
homomorphism. It is easy to see that two homotopic asymptotic homomorphisms
define homotopic asymptotic homomorphisms and that two discretizations
$\{t_i\}_{i\in{\bf N}}$ and $\{t'_i\}_{i\in{\bf N}}$ satisfying the above
properties define homotopic discrete asymptotic homomorphisms too, hence
the map $\Ext^{as}(A,B)\ar\Ext^{as}_{discr}(A,B)$ is well defined. The
inverse map is given by linear interpolation of discrete asymptotic
homomorphisms.
\q
Let $(\varphi_n)_{n\in{\bf N}}$ be a discrete asymptotic homomorphism and
let $(m_n)_{n\in{\bf N}}$ be a sequence of numbers $m_n\in{\bf N}$. Then
we call the sequence
$$
(\underbrace{\varphi_1,\ldots,\varphi_1}_{m_1\ {\rm times}},
\underbrace{\varphi_2,\ldots,\varphi_2}_{m_2\ {\rm times}},
\varphi_3,\ldots)
$$
a {\it reparametrization} of the sequence $(\varphi_n)_{n\in{\bf N}}$.
It is easy to see that a reparametrization does not change the homotopy
class of an asymptotic homomorphism.
\begin{lem}\label{boundedness}
There exists an equicontiuous sequence of liftings $\varphi'_n:A\ar
\M(B\otimes\K)$ for $\varphi_n$.
\end{lem}
\proof
We have already assumed that the sequence $(\varphi_n)_{n\in{\bf N}}$ is
equicontiuous.
By the Bartle--Graves selection theorem \cite{Bartle-Graves}
there exists a continuous selection $s:Q(B\otimes\K)\ar \M(B\otimes\K)$. Put
$\varphi'_n(a)=s\circ\varphi_n(a)$, $a\in A$.
\q
Now we are going to construct the map
$\til{CH}:\Ext^{as}(A,B)\ar[[SA,B\otimes\K]]$.
Due to Lemma \ref{cont=discr} it is
sufficient to define the map $\til{CH}$ as a map from
$\Ext^{as}_{discr}(A,B)$ to $[[SA,B\otimes\K]]_{discr}$.
For $a,b\in A$, $\lambda\in{\bf C}$ put
\begin{eqnarray*}
P_n(a,b)&=&\varphi_n(ab)-\varphi_n(a)\varphi_n(b);\\
L_n(a,b,\lambda)&=&\varphi_n(a+\lambda b)-\varphi_n(a)+\lambda\varphi_n(b);\\
A_n(a)&=&\varphi_n(a^*)-\varphi_n(a)^*
\end{eqnarray*}
and define $P'_n(a,b)$, $L'_n(\lambda,a)$, $A'_n(a)$ in the
same way but with the liftings $\varphi'_n$ instead of $\varphi_n$.
For shortness sake we will write $R_n(a,b,\lambda)$ for either of
$P_n(a,b)$, $L_n(a,b,\lambda)$, $A_n(a)$.
In what follows we identify $B\otimes\K$ (resp. $\M(B\otimes\K)$) with the
$C^*$-algebra of compact (resp. adjointable) operators on the standard
Hilbert $C^*$-module $B\otimes l_2({\bf N})=l_2(B)$ and use the notion
of diagonal operators in this sense. The following lemma
shows how one has to choose a quasicentral approximate unit that makes it
possible to define the map $\til{CH}$.
\begin{lem}\label{qcau}
Let $(\varphi_n)_{n\in{\bf N}}:A\ar Q(B\otimes\K)$ be a discrete
asymptotic homomorphism. Then there exists a reparametrization
of $(\varphi_n)_{n\in{\bf N}}$ and an approximate unit
$(u_n)_{n\in{\bf N}}\subset B\otimes\K$ with the following properties:
\begin{enumerate}
\item
for any $a\in A$ one has
$$
\lim_{n\to\i}\norm{[\varphi'_n(a),u_n]}=0;
$$
\item
for any $\alpha\in C_0(0,1)$, for any $a,b\in A$, $\lambda\in{\bf C}$
one has
$$
\lim_{n\to\i}\norm{\alpha(u_n)P'_n(a,b)}=
\lim_{n\to\i}\norm{\alpha(u_n)L'_n(a,b,\lambda)}=
\lim_{n\to\i}\norm{\alpha(u_n)A'_n(a)}=0;
$$
\item
$\lim_{n\to\i}\norm{u_{n+1}-u_n}=0$;
\item
every $u_n$ is a diagonal operator, $u_n=\diag\{u_n^1,u_n^2,\ldots\}$,
where diagonal entries $u_n^i$ belong to $B$ and
$$
\lim_{i\to\i}\sup_n \norm{u_n^{i+1}-u_n^i}=0.
$$
\end{enumerate}
\end{lem}
\proof
Let $\{F_n\}_{n\in{\bf N}}$ be a generating system for $A$
\cite{Connes-Higson}. This means that every $F_n\subset A$ is
compact,
\ $\ldots\subset F_n\subset F_{n+1}\subset\ldots$\ , \ $\cup_n F_n$ is
dense in $A$ and one has
$$
F_n\cdot F_n\subset F_{n+m(n)};
\quad
F_n+\lambda F_n\subset F_{n+m(n)}, \ \ (|\lambda|\leq 1);
\quad
F_n^*\subset F_{n+m(n)}
$$
for some function $m:{\bf N}\to{\bf N}$.
Put
$$
\e_{n,k}=\sup_{a,b\in F_k,|\lambda|\leq 1}
\max(\norm{P_n(a,b)},\norm{L_n(a,b,\lambda)},\norm{A_n(a)}).
$$
For every fixed $a,b,\lambda$ the sequences $(P_n(a,b))$,
$(L_n(a,b,\lambda))$ and $(A_n(a))$ vanish as $n$
approaches infinity, but the sequence $(\e_{n,n})_{n\in{\bf N}}$
does not have to vanish. Nevertheless
one can reparametrize the sequence $\{F_n\}$ by a sequence
$k=(k_n)_{n\in{\bf N}}$, which approaches infinity slowly enough
and such that $\e_{n,k_n}$ vanishes as $n\to\i$. Put $\e_n=\e_{n,k_n}$.
Then
\be\label{eps}
\lim_{n\to\i}\e_n=0.
\ee
Remind that $R_n(a,b,\lambda)=q(R'_n(a,b,\lambda))$, where
$R'_n(a,b,\lambda)\in \M(B\otimes\K)$ is either of $P'_n(a,b)$,
$L'_n(a,b,\lambda)$, $A'_n(a)$ and $q$ is the quotient map. Define the
compact operators $K_n(a,b,\lambda)\in B\otimes\K$ by the formula
$$
R'_n(a,b,\lambda)=K_n(a,b,\lambda)+s(R_n(a,b,\lambda)),
$$
where $s:Q(B\otimes\K)\ar\M(B\otimes\K)$ is the Bartle--Graves selection
map. Then $K_n(a,b,\lambda)$ are continuous with respect to $a,b,\lambda$
and $\lim_{n\to\i}\norm{s(R_n(a,b,\lambda))}=0$ for every $a,b\in
A$, $\lambda\in{\bf C}$ by Lemma \ref{boundedness}.
Let $e=(e_n)_{n\in{\bf N}}\subset B\otimes\K$ be an approximate unit and
let $Conv(e)$ denote its convex hull.
By induction we can
choose $u_n\in Conv(e)$ in such a way that $u_n\geq u_{n-1}$
and that the estimates
\be\label{1}
\norm{[\varphi'_n(a),u_n]}<\e_n;
\ee
and
\be\label{2}
\norm{u_n K_n(a,b,\lambda)-K_n(a,b,\lambda)}<\e_n
\ee
hold for any $a,b\in F_n$ and any $|\lambda|\leq 1$.
The estimate \ref{2} provides that for any $\alpha\in C_0(0,1)$ and for
every $a,b\in A$, $\lambda\in{\bf C}$ one has
$$
\lim_{n\to\i}\norm{\alpha(u_n)K_n(a,b,\lambda)}=0,
$$
hence
$$
\lim_{n\to\i}\norm{\alpha(u_n)R'_n(a,b,\lambda)}
\leq\lim_{n\to\i}\norm{\alpha(u_n)K_n(a,b,\lambda)}+
\norm{\alpha}\lim_{n\to\i}\norm{s(R_n(a,b,\lambda))}=0,
$$
so the conditions (\ref{1})-(\ref{2}) together with
Lemma \ref{boundedness} provide the first two items of
Lemma \ref{qcau}.
The above choice of $(u_n)_{n\in{\bf N}}$ does not yet ensure the condition
$\lim_{n\to\i}\norm{u_{n+1}-u_n}=0$. To make it hold we have
to renumber the sequence $(\varphi_n)_{n\in{\bf N}}$. At first divide every
segment $[u_n,u_{n+1}]$ into $n$ equal segments
$[u_{n_i},u_{n_{i+1}}]$, $i=1,\ldots,n$. Then as $0\leq u_i\leq 1$ for
all $i$, so we get $\norm{u_{n_{i+1}}-u_{n_i}}\leq\frac{1}{n}$. Finally we
have to change the sequences $(\varphi_1,\varphi_2,\varphi_3,\ldots)$ and
$(u_1,u_2,u_3,\ldots)$ by the sequence
$(\varphi_1,\varphi_2,\varphi_2,\varphi_3,\ldots)$, where each $\varphi_n$
is repeated $n$ times, and by the sequence
$(u_{1_1},u_{2_1},u_{2_2},u_{3_1},u_{3_2},u_{3_3},u_{4_1},\ldots)$
respectively.
To prove the last item of Lemma \ref{qcau} remind that
an approximate unit $e=(e_n)_{n\in{\bf N}}\subset B\otimes\K$ can be
chosen to be diagonal, $e_n=b_n\otimes\epsilon_n$, where
$(b_n)_{n\in{\bf N}}\subset B$ and
$(\epsilon_n)_{n\in{\bf N}}\subset\K$ are approximate units in $B$
and in $\K$ respectively,
so the quasicentral approximate unit $(u_n)_{n\in{\bf N}}\subset
Conv(e)$ can be made diagonal as well, with diagonal entries from $B$.
Let $T$ be the right shift on the standard Hilbert $C^*$-module
$l_2(B)=B\otimes l_2({\bf N})$, $T\in \M(\K)\subset \M(B\otimes\K)$.
We can join $T$ to the
sets $\varphi'_n(F_n)$ in (\ref{1}) when constructing the sequence
$(u_n)$. Then the sequence $[T,u_n]\in B\otimes\K$
would vanish as $n$ approaches infinity. Hence
\be\label{est11}
\lim_{n\to\i}\sup_i \norm{u_n^{i+1}-u_n^i}=0
\ee
and the operators
$$
\diag\{u_n^2-u_n^1,u_n^3-u_n^2,u_n^4-u_n^3,\ldots\}
$$
are compact, so $\lim_{i\to\i}\norm{u_n^{i+1}-u_n^i}=0$.
Take $\e>0$. By (\ref{est11}) there exists some $N$ such that for all $n>N$
one has $\sup_i\norm{u_n^{i+1}-u_n^i}<\e$. Now consider the finite number of
compact operators
$$
\diag\{u_n^2-u_n^1,u_n^3-u_n^2,u_n^4-u_n^3,\ldots\},\quad
1\leq n\leq N.
$$
Due to their compactness there exists some $I$ such that
for $i>I$ one has $\norm{u_n^{i+1}-u_n^i}<\e$ for $1\leq n\leq N$.
Therefore for $i>I$ we have $\norm{u_n^{i+1}-u_n^i}<\e$ for every $n$,
i.e. $\sup_n\norm{u_n^{i+1}-u_n^i}<\e$.
\q
Put now
$$
\til{CH}(\varphi)_n(\alpha\otimes a)=\alpha(u_n)\varphi'_n(a),
\qquad \alpha\in C_0(0,1), \ a\in A,
$$
where $(u_n)_{n\in{\bf N}}$ satisfies the conditions of Lemma \ref{qcau}.
Items $i)-iii)$ of Lemma \ref{qcau} ensure that
$(\til{CH}(\varphi)_n)_{n\in{\bf N}}$
is a discrete asymptotic homomorphism from $SA$ to $B\otimes\K$. If
$(u_n)_{n\in{\bf N}}$ and $(v_n)_{n\in{\bf N}}$ are two quasicentral
approximate unities satisfying Lemma \ref{qcau} and the estimate(\ref{2})
then the linear homotopy $(tu_n+(1-t)v_n)_{n\in{\bf N}}$
provides that the maps $\til{CH}$ defined with the use of these
approximate unities are homotopic. Finally, if $\varphi$ and $\psi$
represent the same homotopy class in $\Ext^{as}_{discr}(A,B)$ then
$\til{CH}(\varphi)$ and $\til{CH}(\psi)$ are homotopic.
If all $\varphi_n$ are constant, $\varphi_n=f:A\ar Q(B\otimes\K)$ with $f$
being a
genuine homomorphism, then obviously $CH(f)=\til{CH}(\varphi)$, so we have
\begin{lem}
The map $\til{CH}:\Ext^{as}(A,B)\ar [[SA,B\otimes\K]]$ is well defined and
the diagram
$$
\diagram
\Ext(A,B)\rto^i\drto_{CH}&\Ext^{as}(A,B)\dto^{\til{CH}}\\
&[[SA,B\otimes\K]].&
\enddiagram
%
%\begin{array}{ccc}
%\Ext(A,B)\!\!\!\!\!&\stackrel{i}{\ar}&\!\!\!\!\!\Ext^{as}(A,B)\\
%&\!\!\!\!\!\!\!\!\!\!{\scriptstyle CH}\searrow&
%\downarrow{\scriptstyle \til{CH}}\\
%&&\!\!\!\!\![[SA,B\otimes\K]].
%\end{array}
$$
is commutative. \q
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\protect\small\protect
\section{An inverse for $\til{CH}$}
}
Let $\alpha_0=e^{2\pi ix}-1$ be a generator for $C_0(0,1)$ and let $T$ be
the right shift on the Hilbert space $l_2({\bf N})$.
Define a homomorphism
$$
g:C_0(0,1)\ar Q(\K) \quad{\rm by}\quad
g(\alpha_0)=q(T)-1,
$$
where $q:\M(B\otimes\K)\ar Q(B\otimes\K)$ denotes the quotient map
Denote by
$$
\iota:Q(\K)\otimes B\otimes\K\subset
Q(B\otimes\K\otimes\K)\cong Q(B\otimes\K)
$$
the standard inclusion and put
$$
j=\iota\circ(g\otimes\id_{B\otimes\K}):
SB\otimes\K\ar Q(\K)\otimes B\otimes\K\ar Q(B\otimes\K).
$$
The homomorphism $j$ obviously induces a map
$$
j_*:[[A,SB\otimes\K]]\ar \Ext^{as}(A,B).
$$
Let $S:[[A,B\otimes\K]]\ar[[SA,SB\otimes\K]]$ denote the suspension map.
Then the composition $M=j_*\circ S$ gives a map
$$
M:[[A,B\otimes\K]]\ar \Ext^{as}(SA,B).
$$
Let
\be\label{beta}
\beta=(\beta_n)_{n\in{\bf N}}:C_0({\bf R}^2)\ar \K
\ee
be a discrete asymptotic homomorphism representing a generator of
$[[C_0({\bf R}^2),\K]]$.
For a discrete asymptotic extension $\varphi=(\varphi_n)_{n\in{\bf
N}}:A\ar Q(B\otimes\K)$ consider its tensor product by $\beta$
$$
\varphi\otimes\beta=
(\varphi_n\otimes\beta_n)_{n\in{\bf N}}:S^2A\ar Q(B\otimes\K)\otimes\K
$$
and denote its composition with the standard inclusion
$Q(B\otimes\K)\otimes\K\subset Q(B\otimes\K)$ by
$$
Bott_1=\iota\circ(\varphi\otimes\beta):
\Ext^{as}(A,B)\ar\Ext^{as}(S^2A,B).
$$
In a similar way define a map
$$
Bott_2:[[A,B\otimes\K]]\ar [[S^2A,B\otimes\K]].
$$
\begin{thm}\label{identity}
One has
$$
M\circ\til{CH}=Bott_1;\quad
\til{CH}\circ M=Bott_2.
$$
\end{thm}
\proof
We start with $M\circ\til{CH}=Bott_1$. Let $H$ be the standard Hilbert
$C^*$-module over $B$, $H=B\otimes l_2({\bf N})$. Put $\H=\oplus_{n\in
{\bf N}}H_n$, where every $H_n$ is a copy of $H$. We identify the
$C^*$-algebra of compact (resp. adjointable) operators on both $H$ and
$\H$ with $B\otimes\K$ (resp. $\M(B\otimes\K)$).
Instead of writing formulas in $Q(B\otimes\K)$ we
will write them in $\M(B\otimes\K)$ and understand them modulo compacts.
Let $\varphi=(\varphi_n)_{n\in{\bf N}}:A\ar Q(B\otimes\K)$ represent an
element
$[\varphi]\in\Ext^{as}_{discr}(A,B)$ and let $\varphi'_n:A\ar \M(B\otimes\K)$
be liftings for $\varphi_n$ as in Lemma \ref{boundedness}.
If $a_n:H_n\ar H_n$ is a sequence of operators then we write
$(a_1\oplus a_2\oplus a_3\oplus\ldots)$ for their direct sum acting on
$\H=\oplus_{n\in{\bf N}}H_n$. In what follows we use a shortcut
$$
\alpha(u_n)\varphi'_n(a)=a_n.
$$
Let $T$ denote the right shift on $\H$, $T:H_n\ar H_{n+1}$.
Remind that $\alpha_0$ is a generator for $C_0(0,1)$ and that it is
sufficient to define asymptotic homomorphisms on the elements of the form
$\alpha\otimes a\otimes\alpha_0\in S^2A$.
The composition map $(M\circ\til{CH}(\varphi))_n:S^2A\ar Q(B\otimes\K)$
acts by
$$
(M\circ\til{CH}(\varphi))_n(\alpha\otimes a\otimes\alpha_0)
=\Bigl(a_n\oplus a_n\oplus a_n\oplus\ldots\Bigr)(T-1)
$$
modulo compacts on $\H$.
Let
$$
\lambda_n=\Bigl(\lambda_n^1\oplus \lambda_n^2\oplus \lambda_n^3
\oplus\ldots\Bigr)\in \M(B\otimes\K)
$$
be a direct sum of scalar operators,
$\lambda_n^i\in{\bf R}$ ($\lambda_n^i$ acts on $H_i$).
We assume that the numbers $\lambda_n^i$ satisfy the properties
\begin{enumerate}
\item
$\lambda_n^1=0$ and $\lim_{i\to\i}\lambda_n^i=1$ for every $n$;
\item
$\lim_{n\to\i}\sup_i |\lambda_n^{i+1}-\lambda_n^i|=0$;
\item
$\lim_{n\to\i}\sup_i |\lambda_{n+1}^i-\lambda_n^i|=0$.
\end{enumerate}
Let $p$ be a projection onto the first coordinate in
$H=B\otimes l_2({\bf N})$ and let $P=(p\oplus p\oplus p\oplus\ldots)$.
Then $P\lambda_n^i=\diag\{\lambda_n^i,0,0,\ldots\}$ and
the map $\beta_n$ (\ref{beta}) can be written as
$$
\beta_n(\alpha\otimes\alpha_0)=P\cdot\alpha(\lambda_n)\cdot(T-1)
\in 1_B\otimes\K\subset \M(B\otimes\K)
$$
and the map $Bott_1(\varphi):S^2A\ar Q(B\otimes\K)$ can be written in the
form
$$
(Bott_1(\varphi))_n(\alpha\otimes a\otimes\alpha_0)=
\Bigl(\alpha(\lambda_n^1)\varphi'_n(a)\oplus
\alpha(\lambda_n^2)\varphi'_n(a)\oplus
\alpha(\lambda_n^3)\varphi'_n(a)\oplus \ldots\Bigr)
(T-1).
$$
One of the obvious choices for $\lambda_n^i$ is to put
$(\lambda_n^i)_{i\in{\bf N}}=
(0,\frac{1}{n},\frac{2}{n},\ldots,\frac{n-1}{n},1,1,\ldots)$,
but for our purposes
it is better to use another choice. Put $\lambda_n^i=\norm{u_i^n}$ for
all $n$ and $i$ and pay notice to the change of subscripts and
superscripts. Lemma \ref{qcau} ensures that the properties $i)-iii)$ are
satisfied (except $\lambda_n^1=0$, which we can assume without any loss of
generality).
Let ${\bf v}_n^i=\diag\{u_i^n,u_i^n,u_i^n,\ldots\}$ be a diagonal
operator on $H$. Define a new asymptotic homomorphism
$\Phi=(\Phi_n)_{n\in{\bf N}}:S^2A\ar Q(B\otimes\K)$ by the formula
$$
(\Phi)_n(\alpha\otimes a\otimes\alpha_0)=
\Bigl(\alpha({\bf v}_n^1)\varphi'_n(a)\oplus
\alpha({\bf v}_n^2)\varphi'_n(a)\oplus
\alpha({\bf v}_n^3)\varphi'_n(a)\oplus \ldots\Bigr)
(T-1).
$$
It is easy to see that the asymptotic homomorphisms $(\Phi_n)_{n\in{\bf
N}}$ and $((Bott_1(\varphi))_n)_{n\in{\bf N}}$ are homotopic
(when $B$ is unital these maps coincide).
Remind that we had
constructed the diagonal quasicentral approximate unit $u_n$ out of the
approximate unit $e=(e_n)_{n\in{\bf N}}$, which had the form
$e_n=b_n\otimes\epsilon_n$ with both $(b_n)_{n\in{\bf N}}$ and
$(\epsilon_n)_{n\in{\bf N}}$ being approximate unities in $B$ and $\K$
respectively. We can assume that $(\epsilon_n)_{n\in{\bf N}}$ is the
standard approximate unity consisting of projections. Let
$(b_t)_{t\in[1,\i)}$ be a continuous appproximate unit for $B$ such that
in the integer points it coincides with $b_n$. Put also $b_\i=1$.
The needed homotopy,
which connects $u_i^n$ with $\lambda_n^i=\norm{u_i^n}$, can be constructed
if we change the approximate unit $e$ by the continuous path of
approximate units $e^t=(e^t_n)_{n\in{\bf N}}$, $t\in[1,\i]$, defined by
$e^t_n=b_{n+t}\otimes\epsilon_n$ for each $t$ in the construction of
$u_n$ in Lemma \ref{qcau}.
Now we have to connect the asymptotic homomorphisms
$(Bott_1(\varphi)_n)_{n\in{\bf N}}$ and
$(M\circ\til{CH}(\varphi)_n)_{n\in{\bf N}}$
by a homotopy in the class of asymptotic
homomorphisms. In fact we are going to do more and to connect
each of these asymptotic homomorphisms with a genuine homomorphism
$f:S^2A\ar Q(B\otimes\K)$ defined modulo compacts by
$$
f(\alpha\otimes a\otimes\alpha_0)=\Bigl(
a_1\oplus a_2\oplus a_3\oplus\ldots
\Bigr)\cdot(T-1),
$$
$\alpha\in C_0(0,1)$, $a\in A$.
Lemma \ref{qcau} ensures that $f$ is indeed a homomorphism.
At first we connect $f$ with $(M\circ\til{CH}(\varphi))_n$ by a path
$F_n(t)$, $t\in[0,1]$. Let $F_n(1)=(M\circ\til{CH}(\varphi))_n$.
Denote $\alpha(u_n)\varphi'_n(a)$ by $a_n$ and put
(modulo compacts)
$$
F_n\left(\frac{1}{2}\right)
(\alpha\otimes a\otimes\alpha_0)=
\Bigl(
\underbrace{a_n\oplus
\ldots\oplus a_n}_{n\ {\rm times}}\oplus
a_{n+1}\oplus a_{n+1}\oplus a_{n+1}\ldots \Bigr)(T-1),
$$
$$
F_n\left(\frac{1}{3}\right)
(\alpha\otimes a\otimes\alpha_0)=
\Bigl(
\underbrace{a_n\oplus
\ldots\oplus a_n}_{n\ {\rm times}}\oplus
a_{n+1}\oplus a_{n+2}
\oplus a_{n+2}\oplus\ldots \Bigr)(T-1),
$$
etc. Finally put
$$
F_n(0)(\alpha\otimes a\otimes\alpha_0)=
\Bigl(
\underbrace{a_n\oplus
\ldots\oplus a_n}_{n\ {\rm times}}\oplus
a_{n+1}\oplus a_{n+2}
\oplus a_{n+3}\oplus\ldots \Bigr)(T-1)
$$
and connect $F_n(1)$, $F_n(\frac{1}{2})$, $F_n(\frac{1}{3})$, $\ldots$ and
$F_n(0)$ by a piecewise linear path $F_n(t)$, $t\in[0,1]$.
It is easy to see
that for every $t>0$ the sequence $(F_n(t))_{n\in{\bf N}}$ is an
asymptotic homomorphism. And as the maps $F_n(0)$ and $f$ differ by
compacts, so they coincide as homomorphisms into $Q(B\otimes\K)$.
Continuity in $t$
is also easy to check. So the asymptotic homomorphism
$(M\circ\til{CH}(\varphi)_n)_{n\in{\bf N}}$ is homotopic to the
homomorphism $f$.
Now we are going to construct a homotopy $F'_n(t)$, $t\in[0,1]$,
which connects $f$ with $(Bott_1(\varphi)_n)_{n\in{\bf N}}$.
For each $k\in{\bf N}$ consider the following sequence $({\bf
u}_n^k)_{n\in{\bf N}}$ of diagonal operators, each of which acts on the
corresponding copy of $H=H_n$ in their direct sum $\H$:
$$
{\bf u}_n^k=\diag\{u_n^1,u_n^2,\ldots,u_n^{k-1},u_n^k,u_n^k,u_n^k\ldots\}.
$$
Put $u_n(\frac{1}{k})={\bf u}_n^k$, $k\in{\bf N}$, and let $u_n(t)$,
$t\in[0,1]$, be a piecewise linear path interpolating the points
$u_n(1)$, $u_n(\frac{1}{2})$, $u_n(\frac{1}{3})$, \ldots and $u_n(0)=u_n$.
Then we get a strictly
continuous path of operators $u_n(t)$, which gives a homotopy
$$
F'_n(t)(\alpha\otimes a\otimes\alpha_0)
=\Bigl(\underbrace{a_{1,n}(t)\oplus\ldots\oplus
a_{n,n}(t)}_{n\ {\rm times}}\oplus
a_{n+1,n+1}(t)\oplus
a_{n+2,n+2}(t)\oplus\ldots
\Bigr)(T-1),
$$
where $a_{i,n}(t)=\alpha(u_i(t))\varphi'_n(a)$.
As
$$
\Phi_n(\alpha\otimes a\otimes\alpha_0)
=\Bigl(a_{1,n}(1)\oplus a_{2,n}(1)\oplus a_{3,n}(1)\oplus\ldots
\Bigr)(T-1),
$$
so for every $\alpha\otimes a$ one has
$$
\lim_{n\to\i}\norm{F'_n(1)(\alpha\otimes a\otimes\alpha_0)-
\Phi_n(\alpha\otimes a\otimes\alpha_0)}=0,
$$
hence the asymptotic homomorphisms $F'_n(1)$ and $\Phi_n$ are equivalent.
But we already know that $\Phi_n$ is homotopic to $Bott_1(\varphi)_n$.
On the other hand, it is easy to see that $F'_n(0)$
coincides with $f$ modulo compacts, so we can finally conclude that
$M\circ\til{CH}=Bott_1$ up to homotopy.
\smallskip
The second identity of Theorem \ref{identity} is much simpler to prove.
For $\psi=(\psi_n)_{n\in{\bf N}}:A\ar B\otimes\K$ we have (modulo compacts)
$$
M(\psi)_n(\alpha_0\otimes a)=\Bigl(\psi_1(a)\oplus\psi_2(a)
\oplus\psi_3(a)\oplus\ldots\Bigr)(T-1),\qquad a\in A.
$$
But as every $\psi_n(a)\in B\otimes\K$, i.e. is compact,
so when choosing a quasicentral
approximate unit $\{w_n\}_{n\in{\bf N}}$ in the definition of the
map $\til{CH}$ we can take it in the form
$$
w_n=\Bigl({\bf w}_1^{(n)}\oplus{\bf w}_2^{(n)}
\oplus{\bf w}_3^{(n)}\oplus\ldots\Bigr),
$$
where each ${\bf w}_i^{(n)}$ is a finite rank diagonal operator of the form
$$
{\bf w}_i^{(n)}=\diag\{\underbrace{\lambda_i b_n,
\ldots,\lambda_i b_n}_{m_n\ {\rm times}},0,0,\ldots\}
$$
for some numbers $(m_n)_{n\in{\bf N}}$, where $(b_n)_{n\in{\bf N}}$ is
a quasicentral
approximate unit for $B$ and the scalars $\lambda_i$ are defined by
$$
\lambda_i=
\left\lbrace\begin{array}{ll}
\frac{n-i+1}{n} & {\rm for}\ i