Shape theory and extensions of C*-algebras

TL;DR

This paper develops a new pairing between homotopy classes of asymptotic homomorphisms and semi-invertible extensions of C*-algebras, establishing conditions under which semi-invertibility and E-theory equivalence are preserved under shape equivalence.

Contribution

It introduces a pairing that relates asymptotic homomorphisms and extensions, providing criteria for semi-invertibility transfer via shape equivalence of C*-algebras.

Findings

The pairing allows transferring semi-invertibility between shape equivalent algebras.

Shape equivalence ensures the preservation of semi-invertibility of extensions.

Conditions under which E-theory coincides with Ext for one algebra extend to shape equivalent algebras.

Abstract

Let $A$, $A'$ be separable $C^*$-algebras, $B$ a stable $\sigma$-unital $C^*$-algebra. Our main result is the construction of the pairing $[[A',A]]\times\operatorname{Ext}^{-1/2}(A,B)\to\operatorname{Ext}^{-1/2}(A',B)$, where $[[A',A]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A'$ to $A$ and $\operatorname{Ext}^{-1/2}(A,B)$ is the group of semi-invertible extensions of $A$ by $B$. Assume that all extensions of $A$ by $B$ are semi-invertible. Then this pairing allows us to give a condition on $A'$ that provides semi-invertibility of all extensions of $A'$ by $B$. This holds, in particular, if $A$ and $A'$ are shape equivalent. A similar condition implies that if $\operatorname{Ext}^{-1/2}$ coincides with $E$-theory (via the Connes-Higson map) for $A$ then the same holds for $A'$.