WAP-biprojectivity of the enveloping dual Banach algebras

Abstract

In this paper, we introduce a new notion of biprojectivity, called W AP -biprojectivity for F (A), the enveloping dual Banach algebra associated to a Banach algebra A. We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dualFor an infinite commutative compact group G, we show that the convolution Banach algebrawe provide some examples of the enveloping dual Banach algebras and we study their W AP -biprojectivity and Connes amenability.F (A) is Connes amenable if and only if A admits a W AP -virtual diagonal [1, Theorem 6.12].Motivated by these results, first we introduce the notion of W AP -biprojectivity for the enveloping dualBanach algebra associated to a Banach algebra A. Next for a Banach algebra A we investigate the relation between W AP -biprojectivity of F (A) with biprojectivity of A and also for a a dual Banach algebra A we study the relation between W AP -biprojectivity of F (A) with Connes biprojectivity of A. We conclude 2010 Mathematics Subject Classification. Primary 46M10, 46H20 Secondary 46H25, 43A10,.

Amenable-like and Connes Amenable-like Propertiesof Certain Matrix Algebras

“…So M is a dual Banach algebra. On the other hand since M is a reflexive Banach space, W AP (M * ) = M * [22]. Thus the enveloping dual Banach algebra F (M) = M. This point of view shows again M is a dual Banach algebra.…”

“…W AP -biprojectivity is a natural variant of biprojectivity that is suitable notion for the class of enveloping dual Banach algebras [22]. F (A) is called W AP -biprojective if there exists a wk * -wk * continuous A-bimodule morphism ρ :…”

“…A-bimodule map induced by π A : A ⊗A −→ A. It has been showed that for an infinite commutative compact group G, the convolution Banach algebra F (L 2 (G)) is not W APbiprojective [22].…”