Cornel Pasnicu: Inclusions of C*-algebras, 305-319

Abstract:

We introduce three notions of inclusions of C*-algebras: with the ideal property, with the weak ideal property, and with topological dimension zero. We characterize these notions and we show that for an inclusion of C*-algebras, the ideal property $\Rightarrow$ the weak ideal property $\Rightarrow$ topological dimension zero. We prove that any two of these three notions do not coincide in general, but they are all equivalent in many interesting cases. We show some permanence properties for these notions, and we prove that they behave well with respect to tensor products and crossed products by discrete (finite) groups, in many interesting cases. For example, we prove that if $A \subseteq B$ is an inclusion of C*-algebras which has topological dimension zero and $\alpha \colon G \to$ Aut$(B)$ is a strongly pointwise outer action of a finite group $G$ on $B$ and if $A$ is $\alpha$-invariant, then the inclusion of crossed products $C^*(G, A, \alpha) \subseteq C^*(G, B, \alpha)$ has topological dimension zero. We show that for an inclusion of C*-algebras, the real rank zero (in the sense of Gabe and Neagu [5]) $\Rightarrow$ the ideal property, and that these two notions do not coincide in general.

Key Words: Inclusions of C*-algebras, ideal property, weak ideal property, topological dimension zero, tensor products of C*-algebras, crossed products, primitive spectrum.

2020 Mathematics Subject Classification: Primary 46L05; Secondary 46L06, 46L55.

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