Asymptotic homomorphisms into the Calkin algebra

TL;DR

This paper investigates the relationship between asymptotic and genuine homomorphisms from a separable C*-algebra to the corona algebra, establishing conditions under which they are equivalent, especially for suspensions.

Contribution

It proves that for suspensions, asymptotic homomorphisms coincide with E-theory and every asymptotic homomorphism is homotopic to a genuine one.

Findings

ext{For suspensions, } ext{Ext}^{as}(A,B) ext{ equals E-theory.

ext{The natural map } i ext{ is surjective.

ext{Any asymptotic homomorphism from } SA ext{ to } M(B)/B ext{ is homotopic to a genuine homomorphism.}

Abstract

Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-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 corona algebra $M(B)/B$ 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 surjective. In particular any asymptotic homomorphism from $SA$ to $M(B)/B$ is homotopic to some genuine homomorphism.