Abstract. We reformulate the Baum-Connes conjecture with coe cients by introducing a new crossed product functor for C⇤-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to...

let $a$ be an arbitrary banach algebra and $varphi$ a homomorphism from $a$ onto $bbb c$. our first purpose in this paper is to give some equivalent conditions under which guarantees a $varphi$-mean of norm one. then we find some conditions under which there exists a $varphi$-mean in the weak$^*$ cluster of ${ain a; |a|=varphi(a)=1}$ in $a^{**}$.

For horocyclic products of percolation subtrees of regular trees, we show almost sure amenability. Under a symmetry condition concerning the growth of the two percolation trees, we show the existence of an increasing Følner sequence (which we call strong amenability).

Abstract Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure crossed product over Fourier algebra acting group. The resulting characterization injectivity for products generalizes a result Anantharaman-Delaroche discrete groups. $C^*$ -algebras in same way, and amenability action is related to nuclearity corresponding pr...

A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions. 1. Introduction. It has been demonstrated in [8], [9] that the Hopf-cyclic homology developed by Connes and Moscovici [5]...