Haagerup Tensor Product and Strong Inequalities with Property for C^*- algebras

Abstract

We consider the natural contractive map from the central Haagerup tensor product of a unital C^*-algebra with itself to the space of completely bounded maps on the C^*-algebra ,and we show a Strong Haagerup inequality with operator coefficients. If for an integer , we costruct the subspace of the von Neumann algebra of a free group spanned by the words of the length given in the generators , then we need to provide an explicit upper bound on the norm on the matrix of the sub spaces of the von Neumann algebra , which improves and generalizes previous results by Kemp–Speicher and Buchholz and Parcet–Pisier. We define the Haagerup property for C^*-algebras and extend this to a notion of relative Haagerup property for the inclusion, on a C^*-subalgebra. We study the Haagerup property for C^*-algebras .We first give new examples of C^*-algebras with the Haagerup property .A nuclear C^*-algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup property for C^*-algebras is established.