A generalized type semigroup and dynamical comparison

Abstract

In this paper, we construct and study a semigroup associated to an action of a countable discrete group on a compact Hausdorff space, that can be regarded as a higher dimensional generalization of the type semigroup. We study when this semigroup is almost unperforated. This leads to a new characterization of dynamical comparison and thus answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither necessarily the acting group is amenable nor the action is minimal.

A categorical study on the generalized type semigroup

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In this short note, we show that the generalized type semigroup W(X, Γ) introduced by the author in [4] belongs to the category W. In particular, we demonstrate that W(X, Γ) satisfies axioms (W1)-( W4) and (W6). When X is a zero-dimensional, we also establish (W5) for the semigroup.

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“…Note that the relation in Definition 1.1 is defined on K(X, Γ). The following simple but fundamental result was established in [4]. Thus W(X, Γ) is a well-defined commutative partially ordered semigroup.…”

“…Suppose that α : Γ X is a continuous action of a countably infinite discrete group Γ on a locally compact Hausdorff second countable space X. The generalized type semigroup W(X, Γ) was introduced in [4] to study the almost unperforation form of the dynamical comparison, as a dynamical analogue of the strict comparison in the C * -setting, which was first introduced by Winter in 2012 and refined by Kerr in [3] to study regularity properties of the reduced crossed product C * -algebra C 0 (X) ⋊ r Γ.…”

“…We begin with recalling the definition of W(X, Γ). We remark that, unlike the original definition in [4], we allow our space X to be locally compact instead of only being compact. Such a version has appeared in [5] written in a language of étale groupoid.…”

Purely infinite locally compact Hausdorff étale groupoids and their $C^*$-algebras

“…Then in [15], the author studied dynamical comparison and introduced paradoxical comparison for actions of non-amenable groups on compact Hausdorff spaces to obtain several dynamical criteria establishing the pure infiniteness for unital reduced crossed product C * -algebras. In [16], the author introduced a new semigroup, called the generalized type semigroup, as a generalization of the type semigroup dating back to Tarski, to study dynamical comparison. It was shown in [16] that the dynamical comparison relates to the almost unperforation of the generalized type semigroup.…”