Asymptotic homomorphisms of C^*-algebras and C^*-extensions
by
Vladimir Manuilov
Moscow State University

Let K be the C^*-algebra of compact operators. An extension of a C^*-algebra A by K is a C^*-algebra E containing K as an ideal, such that E/K=A. If Q denotes the Calkin algebra, i.e. the quotient C^*-algebra of all bounded operators by compacts, Q=M(K)/K, then C^*-extensions of A are in one-to-one correspondence with monomorphisms A --> Q by the Busby invariant. Since 70-s (BDF-theory) it became clear that the classification problem for C^*-extensions is closely related to algebraic topology.

An asymptotic homomorphism j from A to B is a family j_t:A --> B of maps that are only asymptotically (as t --> \infty) additive and multiplicative. Study of such maps began by Connes and Higson in 1990. Their construction produces asymptotic homomorphisms out of C^*-extensions. Various reasons show that the natural equivalence relation for asymptotic homomorphisms is homotopy. The set [[SA, B]] of homotopy classes of asymptotic homomorphisms from the suspension SA=C₀(0, 1)\otimesA of A to B\otimesK is the E-theory group.

Examples show that usually there is much more asymptotic homomorphisms than genuine ones. Nevertheless when one considers asymptotic homomorphisms into the Calkin algebra it turns out that every asymptotic homomorphism is homotopic to a genuine one.

For C^*-extensions homotopy equivalence is closely related to various algebraic equivalences (we will discuss in detail different versions of such equivalences). The corresponding functor of extensions of A by B\otimesK we denote by Ext(A, B). Our main result (obtained jointly with K. Thomsen) is that the Connes-Higson map from Ext(A, B) to [[SA, B]] is an isomorphism when A is a suspension. As a corollary one obtains Bott periodicity and composition (similar to Kasparov's product-intersection in KK-theory) for the Ext groups. One of the basic ingredients of the proof is the notion of asymptotically split extension, i.e. a homomorphism f:A --> Q that has an asymptotic homomorphism j_t:A --> M(K) as a lifting.

Date received: April 24, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagt-29.