A note on crossed products

A Note on Approximation Properties of Crossed Products by Hilbert Bimodules

Let X be a Hilbert bimodule over a C∗-algebra A and OX = A⋊X Z. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on sξs ∗ η as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra OX inherits any standard approximation property such as nuclearit...

Crossed Products by a Coalgebra

We introduce the notion of a crossed product of an algebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimens...

A Note on Rigidity for Crossed Product Von Neumann Algebras

In this note, we will point out, as a corollary of Popa’s rigidity theory, that the crossed product von Neumann algebras for Bernoulli shifts cannot have relative property T. This is an operator algebra analogue of the theorem shown by Neuhauser and Cherix-Martin-Valette for discrete groups. Our proof is different from that for groups.

Note on Cup-products

Skew Products and Crossed Products by Coactions

Given a labeling c of the edges of a directed graph E by elements of a discrete group G, one can form a skew-product graph E ×c G. We show, using the universal properties of the various constructions involved, that there is a coaction δ of G on C∗(E) such that C∗(E ×c G) is isomorphic to the crossed product C ∗(E) ×δ G. This isomorphism is equivariant for the dual action δ̂ and a natural action ...