In this note, we give a short proof of the theorem due to Connes, Feldman, and Weiss that a countable Borel equivalence relation on a standard probably space (X,μ) is μ-hyperfinite if and only if it is μ-amenable.

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...

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L (G), and the Fourier algebra, A(G), of a locally compact group, G. Barry Johnson introduced the important concept of amenability for Banach algebras in [20], where he proved, among many other things, that a group algebra L1(G) is amenable precisely when the locally ...

We prove that any free cocycle action of a countable amenable group $\Gamma$ on II$_1$ factor $N$ can be perturbed by inner automorphisms to genuine action. This {\em vanishing cohomology} property, we call $\mathcal V\mathcal C$, is also closed products with amalgamation over finite groups. But beyond this no other examples C$-groups are known. In turn, considering special actions $\Gamma \cur...

In earlier work the authors introduced dynamic asymptotic dimension, a notion of dimension for topological dynamical systems that applies to many interesting examples. In this paper, we use finiteness of dynamic asymptotic dimension to get information on the K-theory of the associated crossed product C ̊algebras: specifically, we give a new proof of the Baum-Connes conjecture for such actions. T...

Ideal Connes-amenability of dual Banach algebras was investigated in [17] by A. Minapoor, A. Bodaghi and D. Ebrahimi Bagha. They studied weak∗continuous derivations from dual Banach algebras into their weak∗-closed two- sided ideals. This work considers weak∗-continuous derivations of dual triangular Banach algebras into their weak∗-closed two- sided ideals . We investigate when weak∗continuous...

 We study the notion of bounded approximate Connes-amenability for‎ ‎dual Banach algebras and characterize this type of algebras in terms‎ ‎of approximate diagonals‎. ‎We show that bounded approximate‎ ‎Connes-amenability of dual Banach algebras forces them to be unital‎. ‎For a separable dual Banach algebra‎, ‎we prove that bounded‎ ‎approximate Connes-amenability implies sequential approximat...

We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank ≤ 2 · dimH . As an application, we o...

For a large class of word hyperbolic groups Γ the cross product C∗-algebras C(∂Γ)⋊Γ, where ∂Γ denotes the Gromov boundary of Γ satisfy Poincaré duality in K-theory. This class strictly contains fundamental groups of compact, negatively curved manifolds. We discuss the general notion of Poincaré duality for C∗-algebras, construct the fundamental classes for the aforementioned algebras, and prove...