Continuous C*-algebras over topological spaces
Mitsuharu Takeori

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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Advanced Banach Space Theory
