as (G,G)-bimodules, and where the sum is over all irreducible representations of G. Let H be a Hopf algebra. If H is finite dimensional, then H∗ is also a Hopf algebra by dualizing all operations from H. We run into issues if H is infinite-dimensional, but we can find a fix. For a finite-dimensional H-module V , an element v ∈ V , and f ∈ V ∗, define a linear functional on H by cf,v(u) = f(uv)....

We give a solution, via operator spaces, of an old problem in the Morita equivalence of C*-algebras. Namely, we show that C*-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (=...

Given a conditionally completely positive map L on a unital ∗-algebra A, we find an interesting connection between the second Hochschild cohomology of A with coefficients in the bimodule EL = Ba(A⊕M) of adjointable maps, where M is the GNS bimodule of L, and the possibility of constructing a quantum random walk (in the sense of [2, 11, 13, 16]) corresponding to L.

It is an open problem as to whether any bimodule inducing a Morita auto-equivalence of block must have endopermutation source. We prove that, for blocks b with normal defect groups in odd characteristic, stronger result holds, namely that all such bimodules linear also the analogous characteristic 2, provided group specific, slightly restrictive, form.

Let A be a Banach algebra and X a Banach A-bimodule, the derivation D : A → X is semi-inner if there are ξ, μ ∈ X such that D(a) = a.ξ − μ.a, (a ∈ A). A is called semi-amenable if every derivation D : A → X∗ is semi-inner. The dual Banach algebra A is Connes semi-amenable (resp. approximately semi-amenable) if, every D ∈ Z1w _ (A,X), for each normal, dual Banach A-bimodule X, is semi -inner (re...