Approximate Connes-amenability of dual Banach algebras

Abstract

We introduce the notions of approximate Connes-amenability and approximate strong Connes-amenability for dual Banach algebras. Then we characterize these two types of algebras in terms of approximate normal virtual diagonals and approximate σW C−virtual diagonals. We investigate these properties for von Neumann algebras and measure algebras of locally compact groups. In particular we show that a von Neumann algebra is approximately Connes-amenable if and only if it has an approximate normal virtual diagonal. This is the "approximate" analog of the main result of Effros in [E. G. Effros, Amenability and virtual diagonals for von Neumann algebras, J. Funct. Anal. 78 (1988), 137-153].We show that in general the concepts of approximate Connes-ameanbility and Connes-ameanbility are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of A * * implies approximate Connes-amenability or approximate amenability of A are also discussed.2010 Mathematics Subject Classification. Primary 46H25, 46H20; Secondary 46H35.

A generalization on character Connes-amenability

“…For a dual Banach algebra A, if A has a (approximate) normal virtual diagonal, then it is (approximately) Connes-amenable (see [8,Theorem 3.1] and [18,Theorem 4.4.15] for more details). The converse of the mentioned results is no longer valid for the Connes-amenability case [22].…”

“…They characterized the structure of approximately amenable Banach algebras through several different ways. After that, this notion was generalized for dual Banach algebras, namely, approximate Connes-amenability [8] and ideal Connes-amenability [15].…”

“…Motivated to [8,13,[23][24][25], in this paper we introduce and study the concepts of left ϕ-approximate Connes-amenability and left character approximate Connes-amenability for dual Banach algebras where ϕ ∈ σ ω * (A). We firstly investigate the basic properties of left ϕ-approximately Connes-amenable dual Banach algebra.…”

Johnson pseudo-Connes amenability of dual Banach algebras

“…A dual Banach algebra A is called Connes amenable if and only if A has an σwc-virtual diagonal, that is, there exists an element M ∈ (σwc(A ⊗A) * ) * such that a • M = M • a and aπ σwc (M) = a for every a ∈ A [13]. Some new generalizations of Connes amenability like approximate Connes amenability and pseudo-Connes amenability have been introduced by Esslamzadeh et al [2] and Mahmoodi [7]. A unital dual Banach algebra A is approximate Connes amenable if and only if there exists a net (M α ) in (σwc Theorem 3.3].…”

Pointwise amenability for dual Banach algebras

“…The concept of w * -approximately Connes amenability introduced by the first author in [10]. One may see also [5,6,9,11,13,14], for more information on Connes amenability and other related notions.…”