Let S be an inverse semigroup with the set of idempotents E and S/ ≈ be an appropriate group homomorphic image of S. In this paper we find a one-to-one correspondence between two cohomology groups of the group algebra `1(S) and the semigroup algebra `1(S/ ≈) with coefficients in the same space. As a consequence, we prove that S is amenable if and only if S/ ≈ is amenable. This could be consider...

We formulate two new criteria of amenability of discrete equivalence relations: in terms of asymptotically invariant families of leafwise probability measures and in terms of isoperimetric properties of leafwise graph structures. These criteria lead to a geometric proof of the Connes{Feldman{Weiss theorem on coincidence of amenability and hyperrniteness for equivalence relations. Nous consid er...

We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability hyperfiniteness coincide. In this paper we define uniform version graphed bounded vertex degrees prove these two coincide as well. Roughly speaking, a measured graph $\mathcal {G}$ is uniformly hyperfinite if any ${\varepsilon }>0$ there exists $K\geq 1$ such not only {G...