A topology on E-theory

TL;DR

This paper introduces a new topology on the set of homotopy classes of asymptotic morphisms between separable C*-algebras, enriching E-theory with topological structure and establishing continuity properties.

Contribution

It defines a topology on E-theory groups, relates it to the shape category, and introduces the Hausdorffized E-theory group with continuity results.

Findings

A topology on $[[A, B]]$ for separable C*-algebras is established.

The Hausdorffization of the asymptotic category is shown to be equivalent to the shape category.

Continuity of the functor $EL(\, ext{·}\, , B)$ is proved, leading to new results for $KL(\, ext{·}\, , B)$.

Abstract

For separable $C^*$-algebras $A$ and $B$, we define a topology on the set $[[A, B]]$ consisting of homotopy classes of asymptotic morphisms from $A$ to $B$. This gives an enrichment of the Connes--Higson asymptotic category over topological spaces. We show that the Hausdorffization of this category is equivalent to the shape category of Dadarlat. As an application, we obtain a topology on the $E$-theory group $E(A, B)$ with properties analogous to those of the topology on $KK(A, B)$. The Hausdorffized $E$-theory group $EL(A, B) = E(A, B) / \overline{\{0\}}$ is also introduced and studied. We obtain a continuity result for the functor $EL(\,\cdot\,,B)$, which implies a new continuity result for the functor $KL(\,\cdot\,,B)$.