On the Haagerup Property for Partial Crossed Products

Let ( A , G , α ) (A,G,α ) left parenthesis upper A comma upper G comma alpha right parenthesis be a partial dynamical system and let A ⋊ α , r G A α ,r G upper A right normal factor semidirect product Underscript alpha comma r Endscripts upper G denote the associated reduced partial crossed product. We introduce the Haagerup property for partial actions of discrete groups on C ∗ C^* upper C Superscript asterisk -algebras. We prove that the partial crossed product A ⋊ α , r G A α ,r G upper A right normal factor semidirect product Underscript alpha comma r Endscripts upper G has the Haagerup property if and only if both A and the partial action α α 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 ∗ C^* upper C Superscript asterisk -algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits of C ∗ C^* upper C Superscript asterisk -algebras and apply this result to study the Haagerup property of inductive limits of partial crossed products.