**-Haagerup Property for C*-Algebras

Extending the notion of Haagerup property for finite von Neumannalgebras to the general von Neumann algebras, the authors define andstudy the $(**)$-Haagerup property for $C^{*}$-algebras in thispaper. They first give an answer to Suzuki''s question (2013), andthen obtain several results of $(**)$-Haagerup property parallel tothose of Haagerup property for $C^{*}$-algebras. It is proved that anuclear unital $C^{*}$-algebra with a faithful tracial state alwayshas the $(**)$-Haagerup property. Some heredity results concerningthe $(**)$-Haagerup property are also proved.