Crossed products by automorphisms of C(X,D)

Abstract: We consider crossed products of the form $C^* \bigl( {\mathbb{Z}}, \, C (X, D), \, \alpha \bigr)$ in which $D$ is simple, $X$ is compact metrizable, $\alpha$ induces a minimal homeomorphism $h \colon X \to X$, and a mild technical assumption holds. In a number of examples inaccessible via methods based on finite Rokhlin dimension, either because $D$ is not ${\mathcal{Z}}$-stable or because $X$ is infinite dimensional, we prove structural properties of the crossed product, such as (tracial) ${\mathcal{Z}}$-stability, stable rank one, real rank zero, and pure infiniteness.

The method is to find a centrally large subalgebra of the crossed product which is a direct limit of ``recursive subhomogeneous algebras over $D$''. With a better understanding of such direct limits, many more examples would become accessible.

This is joint work with Dawn Archey and Julian Buck.

operator algebras

Audience: researchers in the topic

Western Sydney, IPM joint workshop on Operator Algebras

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