Unsuspended Connective $E$-Theory

TL;DR

This paper establishes connective versions of key results in $C^*$-algebra theory, showing equivalences between certain algebraic structures under specific conditions, thereby advancing the understanding of their homotopy properties.

Contribution

It introduces connective versions of classical theorems, linking $u$-equivalence and asymptotic matrix homotopy equivalence for specific $C^*$-algebras.

Findings

Two separable $C^*$-algebras of the form $C_0(X) ensor A$ are $u$-equivalent iff they are asymptotic matrix homotopy equivalent.

Connective versions of results by Shulman and Dadarlat-Loring are proved.

The work extends the understanding of homotopy equivalences in $C^*$-algebra theory.

Abstract

We prove connective versions of results by Shulman [Shu10] and Dadarlat-Loring [DL94]. As a corollary, we see that two separable $C^*$-algebras of the form $C_0(X) \otimes A$, where $X$ is a based, connected, finite CW-complex and $A$ is a unital properly infinite algebra, are $\bu$-equivalent if and only if they are asymptotic matrix homotopy equivalent.