On the Haagerup property for partial crossed products

TL;DR

This paper introduces the Haagerup property for partial group actions on C*-algebras and establishes its equivalence with the property for the associated partial crossed product, linking it to the algebra and group.

Contribution

It defines the Haagerup property for partial actions and proves its equivalence with the property for the crossed product, extending understanding of this property in operator algebras.

Findings

The partial crossed product has the Haagerup property iff both the algebra and the action have it.

The Haagerup property is preserved under inductive limits of partial crossed products.

The paper applies the results to inductive limits, broadening the class of algebras with the Haagerup property.

Abstract

Let $(A,G,\alpha)$ be a partial dynamical system and let $A\rtimes_{\alpha,r} G$ denote the associated reduced partial crossed product. In this article, we introduce the Haagerup property for partial actions of discrete groups on $C^*$-algebras. We prove that the partial crossed product $A\rtimes_{\alpha,r} G$ has the Haagerup property if and only if both $A$ and the partial action $\alpha$ have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product and that of the underlying $C^*$-algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits and apply this result to study the Haagerup property of inductive limits of partial crossed products.