The (**)-Haagerup property for C*-algebras

Abstract

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the (**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki’s question (2013), and then obtain several results of (**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the (**)-Haagerup property. Some heredity results concerning the (**)-Haagerup property are also proved.