Haagerup property of semigroup crossed products by left Ore semigroups

Abstract

We study the Haagerup property of certain semigroup crossed products. Let P be a left Ore semigroup. Then P generates a group G. We assume that there is an action \(\alpha\) of G on a unital \({\rm C}^*\)-algebra A. If A has an \(\alpha\)-invariant state \(\tau\) and \(D^G_P\) has a GP-invariant state, then \(\tau\) induces a state \(\tau'\) on the reduced semigroup crossed product \(A\rtimes_{\alpha,r} P\). If \((A\rtimes_{\alpha,r} P,\tau')\) has the Haagerup property, then both \((A,\tau)\) and G have the Haagerup property. Conversely, the Haagerup property of \((A,\tau)\) implies that of \((A\rtimes_{\alpha,r} P,\tau')\), when G is amenable.