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\date{}
\author{V. Manuilov}\thanks{Partially supported by
RFFI grant No. 02-01-00574 and by HIII\hspace{-2.3ex}%
\rule{1.9ex}{0.07ex}\,-619.2003.01.}
\address{Dept. of Mechanics and Mathematics, Moscow State University,
Leninskie Gory, Moscow, 119992, Russia}
\email{manuilov@mech.math.msu.su}
\title[Equivalence of two approaches to index theory]{Translation
invariant asymptotic homomorphisms: equivalence of two approaches
in index theory}
\subjclass[2000]{Primary 46L80, Secondary 19K56, 58J20}
\keywords{Index, pseudodifferential operator, $C^*$-algebra extension,
Connes--Higson construction, asymptotic homomorphism}
\begin{document}
\begin{abstract}
The algebra $\Psi(M)$ of order zero pseudodifferential
operators on a compact manifold $M$ defines a well-known $C^*$-extension
of the algebra $C(S^*M)$ of continuous functions on the cospherical bundle
$S^*M\subset T^*M$ by the algebra $\K$ of compact operators.
In his proof of the index theorem, Higson defined and used an
asymptotic homomorphism $T$ from $C_0(T^*M)$ to $\K$, which plays the role
of a deformation for the commutative algebra $C_0(T^*M)$.
Similar constructions exist also for operators and
symbols with coefficients in a $C^*$-algebra. We show that the image of
the above extension under the Connes--Higson construction is $T$
and that this extension can be reconstructed out of $T$. This explains, why
the classical approach to index theory coincides with the one based on
asymptotic homomorphisms.
\end{abstract}
\maketitle
\section{Two ways to define index}
The standard way to define the index of a pseudodifferential elliptic
operator on a compact manifold $M$ comes from the short exact sequence
of $C^*$-algebras
\begin{equation}\label{extens}
\begin{xymatrix}{
0\ar[r]& \K\ar[r]& \Psi(M)\ar[r]& C(S^*M)\ar[r]& 0,
}\end{xymatrix}
\end{equation}
where $\K$ is the algebra of compact operators on $L^2(M)$,
$\Psi(M)$ denotes the norm closure of the algebra of order zero
pseudodifferential operators in the algebra of bounded operators
on $L^2(M)$ and $S^*M$ denotes the cospherical bundle, $S^*M=\{(x,\xi)\in
T^*M:|\xi|=1\}$, in the cotangent bundle $T^*M$. If one deals with
operators having coefficients in a $C^*$-algebra $A$ then one has to
tensor the short exact sequence (\ref{extens}) by $A$:
\begin{equation}\label{extensi}
\begin{xymatrix}{
0\ar[r]& \K\otimes A\ar[r]& \Psi_A(M)\ar[r]& C(S^*M;A)\ar[r]& 0,
}\end{xymatrix}
\end{equation}
where $C(X;A)$ denotes the $C^*$-algebra of continuous functions on $X$
taking values in $A$.
The (main) symbol of a pseudodifferential elliptic operator of order zero
is an invertible element in $C(S^*M;A)$ and the $K$-theory boundary
map $\partial:K_1(C(S^*M;A))\to K_0(\K\otimes A)$ maps the symbol to a class in
$K_0(A)$, which is called the index of the operator.
Another approach, suggested by Higson in \cite{Hig}, is based on the
notion of an asymptotic homomorphism \cite{CH}.
Here one starts with a symbol
$\sigma$ of a pseudodifferential operator of order one and constructs
a {\it symbol class} $[a_\sigma]\in K_0(C_0(T^*M))$ (see details in
\cite{Hig}).
Then one constructs an asymptotic homomorphism from $C_0(T^*M)$ to $\K$ as
follows. In the local coordinates $(x,\xi)$ in $U\times\mathbb R^n\subset
TM$ take a smooth function $a(x,\xi)$ with a compact support,
$a\in C_c^\infty(U\otimes\mathbb R^n)$. Then define a
continuous family of operators $T_{a,t}$, $t\in\mathbb R_+=(0,\infty)$,
on $L^2(U)$ by
\begin{equation}\label{asymT}
T_{a,t}f(x)=\int a(x,t^{-1}\xi)e^{ix\xi}\widehat{f}(\xi)d\xi,
\end{equation}
where $\widehat{f}$ is the
Fourier transform for $f$. Fix an atlas $\{U_k\}$ of charts on a compact
manifold $M$ and let $\{\varphi_k\}$ be a smooth partition of unity
subordinate to that atlas. Take also smooth functions $\psi_k$ on $M$ such
that $\supp\psi_k\subset U_k$ and $\psi_k\varphi_k=\varphi_k$ for all $k$.
For $f\in L^2(M)$ put
\begin{equation}\label{asympT}
T_t(a)f=\sum_k T_{\psi_ka,t}(\varphi_kf).
\end{equation}
It is shown in \cite{Hig}, Lemma 8.4, that this family of operators
defines an asymptotic homomorphism $T=(T_t)_{t\in\mathbb R_+}$ from
$C_c^\infty(T^*M)$ to $\K$ (it is also shown in \cite{Hig}, Lemma 8.7,
that if we take another atlas or other
functions $\varphi_k$ and $\psi_k$ then the resulting asymptotic
homomorphism is asymptotically equal to this one). Therefore this
asymptotic homomorphism defines a $*$-homomorphism $\overline{T}$ from
$C_c^\infty(T^*M)$ to
the asymptotic $C^*$-algebra $C_b(\mathbb R_+;\K)/C_0(\mathbb R_+;\K)$,
where $C_b(\mathbb R_+;\K)$ denotes the algebra of bounded continuous
$\K$-valued functions on $\mathbb R_+$ and due to automatic continuity of
$C^*$-algebra
$*$-homomorphisms one can extend $\overline{T}$ to a $*$-homomorphism
$\widehat{T}:C_0(T^*M)\to C_b(\mathbb R_+;\K)/C_0(\mathbb R_+;\K)$.
Applying the Bartle--Graves selection theorem \cite{Loring}, we obtain an
asymptotic homomorphism
$\widetilde{T}=(\widetilde{T}_t)_{t\in\mathbb R_+}:C_0(T^*M)\to\K$, which
is uniformly continuous and
asymptotically equal to $T$ on smooth functions
(i.e. $\lim_{t\to\infty}T_t(a)-\widetilde{T}_t(a)=0$ for any $a\in
C_c^\infty(T^*M)$). Finally the index of the operator with a
symbol $\sigma$
is defined as the class of the image of $[a_\sigma]$
under the map $K_0(C_0(T^*M))\to K_0(\K)$ induced by $\widetilde{T}$.
Once more,
one can tensor everything by $A$ and construct an asymptotic homomorphism
$$
T=(T_t)_{t\in\mathbb R_+}:C_c^\infty(T^*M;A)\to\K\otimes A
$$
and then change it by a uniformly (with respect to $t$) continuous
asymptotic homomorphism extended to $C_0(T^*M;A)$,
\begin{equation}\label{asympTT}
\widetilde{T}=(\widetilde{T}_t)_{t\in\mathbb R_+}:C_0(T^*M;A)\to
\K\otimes A
\end{equation}
(we keep the same notation $T$ and $\widetilde{T}$ for the case
of $A$-valued symbols).
Remark that the asymptotic homomorphism $T$ is translation invariant, i.e.
$T_{ts}(a)=T_t(a_s)$, where $a_s(x,\xi)= a(x,s^{-1}\xi)$, $a\in
C_c^\infty(T^*M;A)$. The asymptotic homomorphism $\widetilde{T}$ enjoys
the property of asymptotic translation invariance, i.e.
$\lim_{t\to\infty}\widetilde{T}_{ts}(a)-\widetilde{T}_t(a_s)=0$ for any
$a\in C_0(T^*M;A)$.
The purpose of the present paper is to explain, why these two approaches
to define the index of elliptic pseudodifferential operators are
equivalent.
\section{The Connes--Higson construction and its inverse}
In the pioneering paper on asymptotic homomorphisms, \cite{CH},
a construction was given, which
transforms $C^*$-algebra extensions into asymptotic homomorphisms.
Given a $C^*$-extension $0\to B\to E\to D\to 0$, one obtains an asymptotic
homomorphism from the suspension $SD=C_0((0,1);D)$ into $B$. One of
the main results of \cite{MT6} was that the asymptotic homomorphisms
obtained via this Connes--Higson construction possess an additional
important property --- translation invariance. In order to make its
description easier we identify suspension $SA$ with $C_0(\mathbb
R_+;D)$ instead of using $(0,1)$. There is a natural action
$\tau$ of $\mathbb R_+$ on itself by multiplication, $\tau_s(x)=xs$,
$s,x\in\mathbb R_+$, which extends to an action on $SD$ by
$\tau_s(f)(x)=f(sx)$, where $f\in SD=C_0(\mathbb R_+;D)$
(in \cite{MT6} the additive structure on
$\mathbb R$ was used instead of the multiplicative structure on $\mathbb
R_+$).
\begin{dfn}
{\rm
An asymptotic
homomorphism $\varphi=(\varphi_t)_{t\in\mathbb R_+}:SD\to B$ is
{\it translation invariant} if
$\varphi_t(\tau_s(f))=\varphi_{ts}(f)$ for any $f\in
SD$ and for any $t,s\in\mathbb R_+$ and if $\lim_{t\to 0}\varphi_t(f)=0$
for any $f\in SD$. It is {\it asymptotically translation invariant} if
$\lim_{t\to\infty}\varphi_t(\tau_s(f))-\varphi_{ts}(f)=0$ for any $f\in
SA$ and for any $t,s\in\mathbb R_+$ and if $\lim_{t\to 0}\varphi_t(f)=0$
for any $f\in SD$.
Two (asymptotically) translation invariant asymptotic homomorphisms
$\varphi^{(0)},\varphi^{(1)}:SD\to B$ are homotopic if there is an
(asymptotically) translation invariant asymptotic homomorphism $\Phi:SD\to
C[0,1]\otimes B$, whose restrictions onto the endpoints of $[0,1]$
coincide with $\varphi^{(0)}$ and $\varphi^{(1)}$ respectively.
}
\end{dfn}
Note that, by passing to spherical coordinates in the fibers, the
suspension $SC(S^*M;A)$ can be identified with the algebra
$C_{00}(T^*M;A)$ of continuous functions on $T^*M$ vanishing both at
infinity and at the zero section and the asymptotic homomorphism
$\widetilde{T}$ (\ref{asympTT}) can be restricted onto $C_{00}(T^*M;A)$.
\begin{lem}
The asymptotic homomorphism $\widetilde{T}:C_{00}(T^*M;A)\to
\K\otimes A$ is
asymptotically translation invariant.
\end{lem}
\begin{proof}
The family of maps (\ref{asympTT}) is obviously asymptotically invariant
under the action of $\mathbb R_+$ and one easily checks that $\lim_{t\to
0}\widetilde{T}_t(a)=0$ for any $a\in C_{00}(T^*M;A)$.
\end{proof}
From now on we assume that $A$ and $D$ are separable and $B$ is stable and
$\sigma$-unital.
Let $\Ext_h(D,B)$ denote the semigroup of homotopy classes of
$C^*$-extensions of $D$ by $B$. In the case, when $D$ is nuclear, this
functor was defined and studied in \cite{BDF}. Let
$[[SD,B]]_{a,\tau}$ denote the semigroup of asymptotically translation
invariant asymptotic homomorphisms from $SD$ to $B$.
Note that there is a forgetful map
\begin{equation}\label{forge}
\begin{xymatrix}{
[[SD,B]]_{a,\tau}\ar[r]&[[SD,B]]
}\end{xymatrix}
\end{equation}
to the group of homotopy
classes of all asymptotic homomorphisms from $SD$ to $B$, which is the
$E$-theory group $E(SD,B)$. The Connes--Higson construction \cite{CH}
defines a map
$$
\begin{xymatrix}{
CH:\Ext_h(D,B)\ar[r]& [[SD,B]].
}\end{xymatrix}
$$
It was shown in
\cite{MT6} that this map factorizes through the map (\ref{forge})
and the modified Connes--Higson construction
\begin{equation}\label{CH}
\begin{xymatrix}{
\widetilde{CH}:\Ext_h(D,B)\ar[r]& [[SD,B]]_{a,\tau}.
}\end{xymatrix}
\end{equation}
The main result of \cite{MT6} is that the map (\ref{CH}) is an
isomorphism.
This was proved by constructing an inverse map
$$
\begin{xymatrix}{
I:[[SD,B]]_{a,\tau}\ar[r]& \Ext_h(D,B).
}\end{xymatrix}
$$
The map $I$ is constructed as follows (see details in \cite{MT6}).
Let $\varphi=(\varphi_t)_{t\in\mathbb R_+}:SD\to B$ be an asymptotically
translation invariant asymptotic homomorphism. Then, by the Bartle--Graves
continuous selection theorem \cite{Loring}, there is an asymptotically
translation invariant asymptotic homomorphism $\widetilde{\varphi}$, which
is asymptotically equal to $\varphi$ and such that the family of maps
$\widetilde{\varphi}_t:SD\to B$ is uniformly continuous.
Let $\gamma_0\in C_0(\mathbb R_+)$ be a (smooth)
function with support in $[1/2,2]$ such that
$\sum_{i\in\mathbb Z}\gamma_i^2=1$, where $\gamma_i=\tau_{2^i}(\gamma_0)$.
Note that $\gamma_i\gamma_j=0$ when $|i-j|\geq 2$.
Let $e_{ij}$ denote the standard elementary operators on the standard
Hilbert $C^*$-module $H_B=l^2(\mathbb Z)\otimes B$.
We identify the algebra of compact (resp.
adjointable) operators on $H_B$ with the $C^*$-algebra $B\otimes\K$ (resp.
the multiplier $C^*$-algebra
$M(B\otimes\K)$) and let
$$
\begin{xymatrix}{
q:M(B\otimes\K)\ar[r]&
Q(B\otimes\K)=M(B\otimes\K)/B\otimes\K
}\end{xymatrix}
$$
be the quotient $*$-homomorphism.
Put, for $a\in D$,
$$
I_0(\varphi)(a)=\sum_{i,j\in\mathbb Z}\widetilde{\varphi}_{2^i}
(\tau_{2^{-i}}(\gamma_i\gamma_j)\otimes a)\otimes e_{ij}\in M(B\otimes\K)
$$
and $I(\varphi)(a)=q(I_0(\varphi)(a))$.
Then the map $I:D\to Q(B\otimes\K)$ is a $*$-homomorphism, so it defines an
extension of $D$ by $B\otimes\K$, being its Busby invariant \cite{Busby}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main result}
Denote by $[\Psi_A(M)]\in\Ext_h(C(S^*M;A),\K\otimes A)$ the homotopy
class of the extension (\ref{extensi}).
\begin{thm}
The image of $[\Psi_A(M)]$ under the Connes--Higson construction coincides
with the homotopy class of the asymptotic homomorphism
$\widetilde{T}$ if $A$ is separable.
\end{thm}
\begin{proof}
Due to \cite{MT6} we do not need to prove that $CH([\Psi_A(M)])$ is
homotopic to $\widetilde{T}$.
It is sufficient to prove instead that $I(\widetilde{T})$ is
homotopic to the Busby invariant of the extension (\ref{extensi}), which
is easier.
In order to construct the Busby invariant for the extension (\ref{extensi})
one can use the same atlas of charts and the same functions $\varphi_k$ and
$\psi_k$ as in the construction of the asymptotic homomorphism $T_t$
(\ref{asympT}).
Let $\theta$ be a smooth cutting function on $[0,\infty)$, which equals 1
outside a compact set and vanishes at the origin and let
$U\subset M$ be a subset diffeomorphic to a domain in
a Euclidean space. In the local coordinates $(x,\xi)$ in
$U\times\mathbb R^n\subset T^*M$ take a smooth function
$a(x,\xi)$ with a compact support with respect to the first coordinate
and order zero homogeneous with respect to the second coordinate.
Let $f$ be an element of the
Hilbert $C^*$-module $L^2(U)\otimes A$ over $A$. Define an operator
$\Op(a)$ on this Hilbert $C^*$-module by
$$
\Op(a)f(x)=\int a(x,\xi)\theta(|\xi|)e^{ix\xi}\widehat{f}(\xi)\,d\xi,
$$
where $\widehat{f}$ is the Fourier transform for $f$.
Then, for a main symbol $a(x,\xi)\in
C^\infty(S^*M;A)$ defined on the whole $M$,
one can construct an operator $\Op(a)$ on the Hilbert $C^*$-module
$L^2(M)\otimes A$ by
$$
\Op(a)(f)=\sum_k\Op(\psi_ka)(\varphi_kf),\qquad
\Op(a)\in M(\K\otimes A).
$$
The map $q\circ\Op:C^\infty(S^*M;A)\to Q(\K\otimes A)$
is a $*$-homomorphism (cf.
\cite{Palais,L-M}), so, due to automatic continuity, it extends to a
$*$-homomorphism $\underline{\Op}:C(S^*M;A)\to Q(\K\otimes A)$. Using the
Bartle--Graves selection theorem one can obtain a continuous homogeneous
lifting $\widetilde{\Op}:C(S^*M;A)\to M(\K\otimes A)$ for
$\underline{\Op}$.
Let $\gamma_0^s$ and $\gamma_{\pm 1}^s$, $s\in (0,1]$,
be smooth functions in $C_0(\mathbb R_+)$ with support in
$[2^{-1/s},2^{1/s}]$ and in $[2^{\pm 1/s-1},2^{\pm 1/s+1}]$ respectively,
such that $\sum_{i\in\mathbb Z}\gamma_i^2=1$, where
$\gamma_{\pm i}^s=\tau_{2^{\pm(i-1)}}(\gamma_{\pm 1}^s)$ for $i>1$.
%Note that $C_{00}(T^*M;A)$ is spanned by elementary tensors of the form
%$\gamma\otimes a$, where $a\in C(S^*M;A)$ and $g\in C_0(\mathbb R_+)$.
Let at first $a\in C^\infty(S^*M;A)$. Define a map from $C^\infty(S^*M;A)$
to $M(\K\otimes A)$ by
$$
\Psi_s(a)=
\sum_{i,j\in\mathbb Z}T_{1}
(\gamma_i^s\gamma_j^s\theta)\otimes a)\otimes e_{ij}
$$
for $s\in(0,1]$
and
$$
\Psi_0(a)=\Op(a)\otimes e_{00}.
$$
Strict continuity of the family $\Psi_s(a)$ at any
$s\in(0,1]$ is obvious, so we have to check it at $s=0$. By construction,
$\gamma_0^s(x)=1$ for $x\in[2^{-1/s+1},2^{1/s-1}]$, hence
$\gamma_i^s$ strictly converges to zero as $s\to 0$ for
any $i\neq 0$, so for any $f\in L^2(U)\otimes A$ in local coordinates one
has
$$
\lim_{s\to 0}\int a(x,\xi)\gamma_i^s(|\xi|)\gamma_j^s(|\xi|)\theta(|\xi|)
e^{ix\xi}\widehat{f}(\xi)\,d\xi=\left\lbrace\begin{array}{cc}
1, &{\rm if\ } i=j=0;\\0,& {\rm otherwise}\end{array}\right.
$$
because $\widehat{f}\in L^2(\mathbb R^n)\otimes A$, hence
\begin{equation}\label{equ1}
\lim_{s\to 0}T_1((\gamma_0^s)^2\theta\otimes a)(f)-\Op(a)(f)=0
\end{equation}
and
\begin{equation}\label{equ2}
\lim_{s\to 0}T_1(\gamma_i^s\gamma_j^s\theta\otimes a)(f)=0
\end{equation}
whenever either $i$ or $j$ differs from zero. Since the set $\Psi_s(a)$
is uniformly bounded for any $a\in C^\infty(S^*M;A)$, it follows from
(\ref{equ1}) and (\ref{equ2}) that
the family of maps $\Psi_s$, $s\in[0,1]$, is strictly continuous
with respect to $s$, hence this family defines a map
$$
\begin{xymatrix}{
\Psi:C^\infty(S^*M;A)\ar[r]& M(C([0,1];\K\otimes A)),
}\end{xymatrix}
$$
which is obviously a $*$-homomorphism modulo the ideal $C([0,1];\K\otimes
A)$. The $*$-homomorphism $q\circ\Psi:C^\infty(S^*M;A)\to
Q(C([0,1];\K\otimes A))$ extends by continuity to a $*$-homomorphism
$\widetilde{\Psi}:C(S^*M;A)\to Q(C([0,1];\K\otimes A))$.
It remains to show that $\widetilde{\Psi}$ is the required homotopy.
One easily sees that $\widetilde{\Psi}_0=q\circ\widetilde{\Op}\oplus 0$,
so one has to check that $\widetilde{\Psi}_1=I(\widetilde{T})$ and it is
sufficient to check the latter equality on
$C^\infty(S^*M;A)$. Since, for big
enough positive $i$, $\gamma_i\gamma_j\theta=\gamma_i\gamma_j$ and since
$$
\lim_{i\to-\infty}T_1(\gamma_i\gamma_j\otimes a)=
\lim_{i\to-\infty}T_1(\gamma_i\gamma_j\theta\otimes a)=0
$$
for any $a\in C^\infty(S^*M;A)$ (for all $j$, because the only non-trivial
values for $j$ are $i$ and $i\pm 1$), so $q\circ \Psi_1=q\circ\Psi'$, where
$$
\Psi'(a)=
\sum_{i,j\in\mathbb Z}T_{1}
(\gamma_i\gamma_j)\otimes a)\otimes e_{ij}.
$$
By properties of asymptotic homomorphisms $T$ and $\widetilde{T}$ one has
$$
\lim_{i\to-\infty}T_1(\gamma_i\gamma_j\otimes a)=0;
$$
$$
\lim_{i\to-\infty}\widetilde{T}_{2^i}(\tau_{2^{-i}}(\gamma_i\gamma_j)\otimes
a)=\lim_{i\to-\infty}
\widetilde{T}_{2^i}(\gamma_0\gamma_{j-i})\otimes a)=0
$$
and
$$
\lim_{i\to\infty}\widetilde{T}_{2^i}(\tau_{2^{-i}}(\gamma_i\gamma_j)\otimes
a)-T_1(\gamma_i\gamma_j\otimes a)
=\lim_{i\to\infty}\widetilde{T}_{2^i}(\gamma_0\gamma_{j-i}\otimes a)-
T_{2^i}(\gamma_0\gamma_{j-i}\otimes a)=0
$$
for any $a\in C^\infty(S^*M;A)$. Therefore,
$\widetilde{\Psi}_1=q\circ\Psi'=I(\widetilde{T})$ on $C^\infty(S^*M;A)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concluding remarks}
So we have proved that the extension (\ref{extensi}) and the
restriction of the asymptotic homomorphism (\ref{asympTT}) to
$C_{00}(S^*M;A)$ define each other. This is the reason beneath the
fact that the two definitions of the index give the same result.
To complete the picture we have to
show this well known fact. For
that purpose consider the extensions
\begin{equation}\label{discs}
\begin{xymatrix}{
0\ar[r]& C_0(T^*(M);A)\ar[r]& C(D^*M;A)\ar[r]& C(S^*M;A)\ar[r]& 0
}\end{xymatrix}
\end{equation}
and
\begin{equation}\label{discs0}
\begin{xymatrix}{
0\ar[r]& C_{00}(T^*(M);A)\ar[r]& C_0(D^*M;A)\ar[r]& C(S^*M;A)\ar[r]& 0,
}\end{xymatrix}
\end{equation}
where $D^*M$ denotes the ball bundle obtained from $T^*M$ by
compactifying each fiber by a sphere
and $C_0(D^*M;A)\subset C(D^*M;A)$ denotes the
subset of functions vanishing on the zero section of $D^*M$.
Consider the diagram
\begin{equation}\label{diagr1}
\begin{xymatrix}{
K_1(C(S^*M;A))\ar[rr]^-{\partial}\ar[d]_-{j}\ar[dr]^-{\partial'}&&
K_0(\K\otimes A)\\
K_0(C_{00}(T^*M;A))\ar[r]^-{i}&
K_0(C_0(T^*M;A))\ar[ur]_-{\widetilde{T}_*}&
}\end{xymatrix}
\end{equation}
where the map $j$ is the standard isomorphism
$K_1(B)=K_0(SB)$, the
map $\partial$ is the $K$-theory
boundary map induced by the extension (\ref{extensi}),
the map $\partial'$ is the $K$-theory boundary map induced by the
extension (\ref{discs}),
the map $\widetilde{T}_*$ is induced by the asymptotic homomorphism
$\widetilde{T}$ (\ref{asympTT}) and the map $i$ is induced by the inclusion
$C_{00}(T^*M)\subset C_0(T^*M)$.
\begin{prop}
The diagram $($\ref{diagr1}$)$ commutes.
\end{prop}
\begin{proof}
It obviously follows from the properties of $E$-theory \cite{CH}
(more details can be found in \cite{CHpreprint}, cf. \cite{blackadar},
Exercise 25.7(d)) that
for any extension $0\to B\to E\to A\to 0$ with the Busby invariant
$\psi:SA\to Q(B)$ the map $K_0(SA)\to K_0(B)$
induced by the asymptotic homomorphism $CH(\psi)$ coinsides with
the $K$-theory boundary map $K_1(A)\to K_0(B)$ after the standard
identification $K_1(A)=K_0(SA)$. Thus $\partial=\widetilde{T}_*\circ i\circ j$.
Since the extension (\ref{discs0}) is the restriction of the extension
(\ref{discs}), we have $\partial'=i\circ j$. Hence the whole diagram commutes.
\end{proof}
%It remains to say that it follows from \cite{Hig} that
%if $\sigma(D)$ is the main symbol of a zero order pseudodifferential
%elliptic operator $D$ then $\delta'([\sigma(D)])$ equals the symbol
%class of the first order operator $D(1+\Delta)^{1/2}$, where $\Delat$
%is the laplacian on $M$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
It may seem that the asymptotic homomorphism
$\widetilde{T}$ (\ref{asympTT}) contains
more information than the extension (\ref{extensi}) since it is defined
not only on $C_{00}(T^*M;A)$, but on the bigger $C^*$-algebra $C_0(T^*M;A)$.
In fact, the extension (\ref{extensi}) also possesses an additional
property, which is equivalent to that additional property of the
asymptotic homomorphism $\widetilde{T}$.
Namely, there is a subalgebra $C(M;A)\subset
C(S^*M;A)$ consisting of functions that are constants on the fibers and
the Busby invariant of the extension (\ref{extensi}) restricted onto
$C(M;A)$ can be lifted to $M(\K\otimes A)$. Indeed, multiplication
$\pi(a)f=af$ for $a\in C(M;A)$ and $f\in L^2(M)\otimes A$ defines such
a lifting, i.e. a $*$-homomorphism $\pi:C(M;A)\to M(\K\otimes A)$.
Using a relative version of the Bartle--Graves theorem
\cite{Loring}, one can construct a continuous section $\overline{\Op}:
C(S^*M;A)\to M(\K\otimes A)$ such that its restriction onto $C(M;A)$
coincides with $\pi$.
So we now describe how one can extend the Connes--Higson
construction to the case, when an extension of a $C^*$-algebra $D$
restricted to a $C^*$-subalgebra $C\subset D$ is liftable.
Denote the Busby invariant of such an extension by $\chi:D\to Q(B)$
and let $\overline{\chi}:D\to M(B)$ be a continuous homogeneous
lifting for $\chi$ such that $\overline{\chi}|_{C}$ is a $*$-homomorphism.
Consider the $C^*$-subalgebra
\begin{equation}\label{alg}
C_0([0,\infty);C)\cup C_0(\mathbb R_+;D)
\end{equation}
in $C_0([0,\infty);D)$.
The Connes--Higson construction on $SD=C_0(\mathbb R_+;D)$ can be defined
on elementary tensors of the form $f\otimes d$, $f\in C_0(\mathbb R_+)$,
$d\in D$, by the formula
\begin{equation}\label{CH1}
CH(\chi)_t(f\otimes d)=\overline{\chi}(d)(f\circ\kappa)(u_t),
\end{equation}
where $(u_t)_{t\in\mathbb R_+}\subset B$ is a quasicentral (with respect
to $\overline{\chi}(D)$) approximate unit, $0\leq u_t\leq 1$, and
$\kappa:(0,1]\to [0,\infty)$ is a homeomorphism (cf. \cite{CH}). In order
to extend this construction to the $C^*$-algebra (\ref{alg}) we have to
define the asymptotic homomorphism $CH(\chi)$ on
$C_0([0,\infty);C)$ compatible with (\ref{CH1}).
Let $g\otimes c\in C_0([0,\infty);C)$ be an elementary tensor,
$g\in C_0[0,\infty)$, $c\in C$. Then apply the same formula,
$$
CH(\chi)_t(g\otimes c)=\overline{\chi}(c)(g\circ\kappa)(u_t).
$$
In the case, when $D=C(S^*M;A)$ and $C=C(M;A)$, the $C^*$-algebra
(\ref{alg}) obviously coincides with the $C^*$-algebra $C_0(T^*M;A)$ and
the extended Connes--Higson construction gives us the asymptotic
homomorphism $\widetilde{T}$ defined on the whole $C_0(T^*M;A)$.
\begin{thebibliography}{99}
\bibitem{blackadar}
{B. Blackadar}, {\it $K$-theory for operator algebras}. MSRI
Publications, {\bf 5}.
Cambridge Univ. Press, 1998.
\bibitem{BDF}
{L. G. Brown, R. G. Douglas, P. A. Fillmore.} {\it Extensions
of $C^*$-algebras and $K$-homology}, Ann. Math. {\bf 105} (1977),
265--324.
\bibitem{Busby}
{R. C. Busby}, {\it Double centralizers and extensions of
$C^*$-algebras},
Trans. Amer. Math. Soc.
{\bf 132} (1968), 79--99.
\bibitem{CH}
{A. Connes, N. Higson},
{\it D\'eformations, morphismes asymptotiques et
$K$-th\'eorie bivariante}, C. R. Acad. Sci. Paris S\'er. I Math. {\bf 311}
(1990), 101--106.
\bibitem{CHpreprint}
{A. Connes, N. Higson}, {\it Almost homomorphisms and KK-theory},
unpublished preprint, http://www.math.psu.edu/higson/Papers/CH1.pdf
\bibitem{Hig}
{N. Higson}, {\it On the $K$-theory proof of the index theorem}, Index
theory and operator algebras (Boulder, CO, 1991), 67--86, Contemp. Math.,
148, Amer. Math. Soc., Providence, RI, 1993.
\bibitem{Loring}
{T. Loring}, {\it Almost multiplicative maps between
$C^*$-algebras}, Operator Algebras and Quantum Field Theory.
Internat. Press, 1997, 111--122.
\bibitem{L-M}
{G. Luke, A. S. Mishchenko}, {\it Vector Bundles and Their Applications},
Kluwer, 1998.
\bibitem{MT4}
{V. Manuilov, K. Thomsen}, {\it $E$-theory is a special case of
$KK$-theory}, Proc. London Math. Soc. {\bf 88} (2004), 455--478.
\bibitem{MT6}
{V. Manuilov, K. Thomsen}, {\it Extensions of $C^*$-algebras and translation
invariant asymptotic homomorphisms}. Preprint.
\bibitem{Palais}
{R. Palais}, {\it Seminar on the Atiyah--Singer index theorem}. Princeton
Univ. Press, 1965.
\end{thebibliography}
\end{document}
\vspace{2cm}
\parbox{7cm}{V. M. Manuilov\\
Dept. of Mech. and Math.,\\
Moscow State University,\\
Moscow, 119992, Russia\\
e-mail: manuilov@mech.math.msu.su
}
\end{document}