Abstract. Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded...

De tels bimodules interviennent dans la théorie cohomologique des algèbres d’opérateurs. Ils sont souvent appelés A − B bimodules d’opérateurs. Ici nous les appellerons plutôt A − B bimodules L matriciellement normés. Etant donné un A−B bimodule normé V , il est naturel de chercher des critères assurant l’existence d’une structure de A−B bimodule L matriciellemnt normé sur V compatible avec la ...

Let A be a dual Banach algebra with predual A∗ and consider the following assertions: (A) A is Connes-amenable; (B) A has a normal, virtual diagonal; (C) A∗ is an injective A-bimodule. For general A, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for A = M(G) where G...

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E p,q -groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and ...

The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a B-A bimodule M which is a generator left B-module is formally smooth if and only if the M-Hochschild dimension of B is at most one. It is also shown that modules M which are generators in the category σ[M ] of M-subgenerated modules provide natural examples of formally...

In the present paper, we give a short proof of the nuclearity property of a class of CuntzPimsner algebras associated with a HilbertA-bimodule M, whereA is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.