Connes Amenability of $l^1$-Munn Algebras

In this paper, we study Connes amenability of $l^1$-Munn algebras. We apply this results to semigroup algebras. We show that for a weakly cancellative semigroup $S$ with finite idempotents, amenability and Connes amenability are equivalent.

Johnson pseudo-Connes amenability of dual Banach algebras

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

Approximate Connes-biprojectivity of dual Banach algebras

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and [Formula: see text]-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connes-biprojective. Next, we show that for a locally compact group [Formula: see text], the Banach algebra [Formula: see text] is approximately Connes-biprojective if and only if [Formula: see text] is amenable. Finally, for an infinite commutative compact group [Formula: see text], we show that the Banach algebra [Formula: see text] with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.

Dual Fréchet algebras: Connes amenability and (σwc)-virtual diagonals

A relation between Connes amenability of dual Banach algebras and the existence of ([Formula: see text])-virtual diagonals was investigated by Volker Runde. In this paper, we first introduce and study the notion of dual Fréchet algebras. We then study the weak[Formula: see text]-weakly continuous elements of Fréchet bimodules, and the notion of normality for dual spaces of Fréchet bimodules. We obtain some important results about these notions. Finally, we introduce and study amenability for dual Fréchet algebras and investigate the relation between this notion and the existence of ([Formula: see text])-virtual diagonals.

A generalization on character Connes-amenability

In the current paper, we introduce the concepts of left ?-approximate Connes-amenability and left character approximate Connes-amenability of a dual Banach algebra A that ? is a ?*-continuous homomorphism fromAto C. We also characterize left ?-approximate Connes-amenability ofAin terms of certain derivations and study some hereditary properties for such Banach algebras. Some examples show that these new notions are different from approximate Connes-amenability and left character Connesamenability for dual Banach algebras.

-CONTRACTIBILITY AND -CONNES AMENABILITY COINCIDE WITH SOME OLDER NOTIONS

It is shown that various definitions of $\unicode[STIX]{x1D711}$-Connes amenability and $\unicode[STIX]{x1D711}$-contractibility are equivalent to older and simpler concepts.