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\sloppy

\date{}

\author{V. Manuilov}\thanks{Partially supported by
RFFI grant No. 05-01-00923 and by Swiss National Science
Foundation.}

\address{V. Manuilov, Dept. of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia
\ {\sl and} \ Harbin Institute of Technology, Harbin, P. R. China}

\email{manuilov@mech.math.msu.su}

\title[On equivalence of two approaches in index theory]{On equivalence of
two approaches in index theory}

\dedicatory{In memory of Yu. P. Solovyov}

\subjclass[2000]{Primary 46L80, Secondary 19K56, 58J20}

\keywords{ $C^*$-algebra extension, Connes--Higson construction,
asymptotic homomorphism, pseudodifferential operator, index}

\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. Recently we have shown that
the image of the above extension under the Connes--Higson construction is
$T$ and that this extension can be reconstructed out of $T$. That is why
the classical approach to the index theory coincides with the one based
on asymptotic homomorphisms. But the image of the above extension is
defined only outside the zero section of $T^*(M)$, so it may seem that
the information encoded in the extension is not the same as that in the
asymptotic homomorphism. We show that this is not the case.

\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}
0\to \K\to \Psi(M)\to C(S^*M)\to 0,
 \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}
0\to \K\otimes A\to \Psi_A(M)\to C(S^*M;A)\to 0,
 \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 $K_1(C(S^*M;A))\to K_0(\K\otimes A)$ 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}. Let us recall the
definition. For $C^*$-algebras $A$ and $B$, a family of maps
$\varphi=(\varphi_t)_{t\in\mathbb R_+}:A\to B$ is called an {\it asymptotic
homomorphism} if the map $t\mapsto \varphi_t(a)$ is continuous for any
$a\in A$ and if this family behaves as a $*$-homomorphism asymptotically
as $t\to\infty$, i.e. $\lim_{t\to\infty}\varphi_t(a+\lambda
b^*)-\varphi_t(a)-\lambda \varphi_t(b)^*=0$ and
$\lim_{t\to\infty}\varphi_t(ab)-\varphi_t(a)\varphi_t(b)=0$ for any
$a,b\in A$, $\lambda\in\mathbb C$.

In the Higson's construction 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, when $A$ is separable.

In \cite{M_JOP} we showed that, for any separable coefficient
$C^*$-algebra $A$, knowing the Higson's asymptotic homomorphism
(\ref{asympTT}), one can reconstruct the corresponding extension and
knowing the extension (\ref{extensi}), one can obtain the corresponding
asymptotic homomorphism except over the zero section of $T^*(M)$. This
suffices to see that the two definitions of index coincide. But it may
seem that asymptotic homomorphisms contain more information, since they
are defined on a bigger set.  The purpose of the present paper is to show
that this is not the case: there is some additional information in
extensions that allows to obtain the whole asymptotic homomorphism and
this information can also be reconstructed back from asymptotic
homomorphisms.

\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 $B$ and 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}
[[SD,B]]_{a,\tau}\to[[SD,B]]
 \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
 $$
CH:\Ext_h(D,B)\to [[SD,B]].
 $$
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}
\widetilde{CH}:\Ext_h(D,B)\to [[SD,B]]_{a,\tau}.
 \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
 $$
I:[[SD,B]]_{a,\tau}\to \Ext_h(D,B).
 $$
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
 $$
q:M(B\otimes\K)\to
Q(B\otimes\K)=M(B\otimes\K)/B\otimes\K
 $$
be the quotient $*$-homomorphism.

Put, for $a\in D$,
 \begin{equation}\label{I_0}
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)
 \end{equation}
and
 \begin{equation}\label{I}
I(\varphi)(a)=q(I_0(\varphi)(a)).
 \end{equation}
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.

%\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}). The following theorem is the main
result of \cite{M_JOP}.

\begin{thm}\label{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}

\section{Relative Connes--Higson construction}

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 the relative version of the Bartle--Graves theorem from
\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 split.

For a pair of $C^*$-algebras $C\subset D$, any extension $0\to B\to E\to
D\to 0$ naturally restricts to an extension $0\to B\to E'\stackrel{p}{\to}
C\to 0$ with $E'\subset E$. The latter extension is {\it split} (resp.
{\it asymptotically split}) if there is a $*$-homomorphism (resp. an
asymptotic homomorphism) $s:C\to E'$ such that $p\circ s=\id_C$. In terms
of Busby invariants this means that if we restrict the Busby invariant
$\chi:D\to Q(B)$ to the $C^*$-subalgebra $C$ then the map $\chi|_C:C\to
Q(B)$ lifts to a $*$-homomorphism (resp. to an asymptotic homomorphism)
from $C$ to $M(B)$. Let $\Ext_h(D,C;B)$ (resp. $\Ext_h^{as}(D,C;B)$)
denote the semigroup of homotopy classes of $C^*$-extensions of $D$ by
$B$, whose restrictions onto $C$ are split (resp. asymptotically split).
There is a tautological map $\Ext_h(D,C;B)\to \Ext_h^{as}(D,C;B)$ and
the extension (\ref{extensi}) defines an element
 $$
[\Psi_A(M)]\in \Ext_h^{(as)}(C(S^*M;A),C(M;A);\K\otimes A).
 $$

For the Busby invariant $\chi:D\to Q(B)$ of the extension
$0\to B\to E\to D\to 0$,
let $\overline{\chi}:D\to M(B)$ be a continuous homogeneous
lifting for $\chi$ such that $\overline{\chi}|_{C}$ is a $*$-homomorphism
(it exists by the relative version of the Bartle--Graves theorem,
\cite{Loring}).
Consider the $C^*$-subalgebra
 \begin{equation}\label{alg}
SD_C=C_0([0,\infty);C)\cup C_0(\mathbb R_+;D)\subset C_0([0,\infty);D).
 \end{equation}
The asymptotic homomorphism $CH(\chi)$ on $SD=C_0(\mathbb R_+;D)$
given by the Connes--Higson construction is 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 $SD_C$
(\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,
 $$
CH(\chi)_t(g\otimes c)=\overline{\chi}(c)(g\circ\kappa)(u_t).
 $$
It is easy to see that thus obtained asymptotic homomorphism on $SD_C$ is
(asymptotically) translation invariant if its restriction onto $SD$ is.
Note that an (asymptotic) translation invariance of
$\varphi=(\varphi_t)_{t\in\mathbb R_+}:SD_C\to B$ does imply that
$\lim_{t\to 0}\varphi_t(f)=0$ only for $f\in C_0(\mathbb R_+;D)$ but not
for $f\in C_0([0,\infty);C)$.

Thus we obtain a semigroup homomorphism
$CH:\Ext_h(D,C;B)\to[[SD_C,B]]_{a,\tau}$ that extends the map (\ref{CH}).

An inverse map $I$ can be defined in the way similar to (\ref{I}). Let
$\varphi=(\varphi_t)_{t\in\mathbb R_+}:SD_C\to B$ be an asymptotically
translation invariant asymptotic homomorphism. Without loss of generality
we may assume that it is uniformly continuous. For $a\in D$, define
$I_0(\varphi)(a)$ and $I(\varphi)(a)$ by the same formulas as
in (\ref{I_0}) and (\ref{I}). Let us show that the map $I(\varphi)$
is asymptotically liftable, when
$a\in C\subset D$. To this end, define the function $\beta\in
C_0[0,\infty)$ by $\beta=1-\sum_{i=0}^\infty\gamma_{-i}$ and put
 \begin{eqnarray*}
I_n(\varphi)(a)&=&\sum_{i,j=-\infty}^n
\varphi_{2^i} (\tau_{2^{-i}}(\gamma_i\gamma_j)\otimes
a)\otimes e_{ij}
+\varphi_{2^n} (\beta^2\otimes a)\otimes a_{n+1,n+1}\\
&+&\varphi_{2^n} (\beta\gamma_0\otimes a)\otimes (e_{n,n+1}+e_{n+1,n})
\in M(B\otimes\K)
 \end{eqnarray*}
It is easy to check that the sequence of maps $I_n(\varphi)$ is a discrete
asymptotic homomorphism, when $a\in C$, and that this sequence can be
interploated to a continuous asymptotic homomorphism $I_t(\varphi)$, $t\in
\mathbb R_+$, from $C$ to $M(B\otimes\K)$ (one has to connect
$\tau(\beta)$ with $0$ by a linear homotopy). It is also obvious that the
values $I_t(\varphi)(a)$ coincide modulo $B\otimes\K$, i.e. that $q\circ
I_t(\varphi)(a)=I(\varphi)(a)$ for any $t\in\mathbb R_+$ and for all $a\in
C$.

So the map $I$ applied to an asymptotic homomorphism $\varphi$
from $[[SD_C,B]]_{a,\tau}$ results in an extension from
$\Ext_h^{as}(D,C;B)$. Thus we get a map
$I:[[SD_C,B]]_{a,\tau}\to\Ext_h^{as}(D,C;B)$ that extends the map (\ref{I})
and, in the same way as in \cite{MT6}, one proves the following statement.

\begin{thm}\label{T4}
The map
 $$
CH:\Ext_h^{as}(D,C;B)\to[[SD_C,B]]_{a,\tau}
 $$
is an isomorphism.

\end{thm}

In the case, when $D=C(S^*M;A)$ and $C=C(M;A)$, the $C^*$-algebra
$SC(S^*M;A)_{C(M;A)}$ (\ref{alg}) obviously coincides with the
$C^*$-algebra $C_0(T^*M;A)$ and we obtain the following statement.

\begin{thm}
The image of $[\Psi_A(M)]\in\Ext_h(C(S^*M;A),C(M;A);\K\otimes A)$ in
$[[C_0(T^*M;A),\K\otimes A]]$ under the Connes--Higson construction
coincides with the asymptotic homomorphism $\widetilde{T}$
$($\ref{asympTT}$)$, when $A$ is separable.

\end{thm}

\begin{proof}
Due to Theorem \ref{T4}, as in the proof of Theorem \ref{THM}, instead of
proving that $CH(\Psi_A(M))=\widetilde{T}$, it is easier to check that
$I(\widetilde{T})=\Psi_A(M)$. The latter equality can be checked by direct
calculation as in the proof of Theorem \ref{THM}, see \cite{M_JOP}.

\end{proof}



\begin{thebibliography}{9}

\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{Hig}
{N. Higson}, {\it On the $K$-theory proof of the index theorem}, Index
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\end{thebibliography}

%\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
%}



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