Continuous C*-algebras over topological spaces

TL;DR

This paper introduces the concept of continuous C*-algebras over topological spaces and proves that such algebras are KK(X)-equivalent to stable Kirchberg algebras under certain conditions, extending known results to a broader setting.

Contribution

It defines continuous C*-algebras over topological spaces and establishes their KK(X)-equivalence to stable Kirchberg algebras, generalizing previous results for single-point spaces.

Findings

Continuous C*-algebras over X are equivalent to C_0(X)-algebras for locally compact Hausdorff X.

Every continuous, full, separable, nuclear C*-algebra over X is KK(X)-equivalent to a stable Kirchberg algebra over X.

Established an X-equivariant exact embedding result for continuous C*-algebras over X.

Abstract

We define continuous C*-algebras over a topological space X and establish some basic results. If X is a locally compact Hausdorff space, continuous C*-algebras over X are equivalent to ordinary continuous C_0(X)-algebras. The main purpose of our study is to prove that every continuous, full, separable, nuclear C*-algebra over X is KK(X)-equivalent to a stable Kirchberg algebra over X. (Here a Kirchberg algebra over X is a separable, nuclear, and strongly purely infinite C*-algebra over X with primitive ideal space homeomorphic to X.) In the case that X is a one-point space, this result is known as that every separable nuclear C*-algebra is KK-equivalent to a stable Kirchberg algebra. Moreover, as an intermediate result, we obtain the X-equivariant exact embedding result for continuous C*-algebras over X.