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We show that Property (TTT) is an obstruction to weak amenability with Cowling–Haagerup constant 1. More precisely, if G is a countable group and H is an infinite subgroup of G such that the pair (G, H) has relative Property (TTT) , then the weak Haagerup constant Λ WH (G) is strictly greater than 1. We apply this result to some semidirect products and lattices in higher rank algebraic groups. [ABSTRACT FROM AUTHOR]

Using Herz-Schur multipliers of a -dynamical system, we introduce the weak Haagerup property of a -dynamical system and prove that it is equivalent to the weak Haagerup property of the associated -crossed product. On the other hand, we investigate an additional assumption under which the weak Haagerup property of the original -algebra and the group implies that of the associated -dynamical system. We also define and study the weak Haagerup property of a -dynamical system. [ABSTRACT FROM AUTHOR]

We show that if $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ has the weak Haagerup property, then both $M$ and $\unicode[STIX]{x1D6E4}$ have the weak Haagerup property, and if $\unicode[STIX]{x1D6E4}$ is an amenable group, then the weak Haagerup property of $M$ implies that of $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$. We also give a condition under which the weak Haagerup property for $M$ and $\unicode[STIX]{x1D6E4}$ implies that of $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$. [ABSTRACT FROM AUTHOR]

We define and study the weak Haagerup property for C*-algebras in this article. A C*-algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group C*-algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of C*-algebras. [ABSTRACT FROM AUTHOR]