In this paper, we introduce a new notion of biprojectivity, called W AP -biprojectivity for F (A), the enveloping dual Banach algebra associated to a Banach algebra A. We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dualFor an infinite commutative compact group G, we show that the convolution Banach algebrawe provide some examples of the enveloping dual Banach algebras and we study their W AP -biprojectivity and Connes amenability.F (A) is Connes amenable if and only if A admits a W AP -virtual diagonal [1, Theorem 6.12].Motivated by these results, first we introduce the notion of W AP -biprojectivity for the enveloping dualBanach algebra associated to a Banach algebra A. Next for a Banach algebra A we investigate the relation between W AP -biprojectivity of F (A) with biprojectivity of A and also for a a dual Banach algebra A we study the relation between W AP -biprojectivity of F (A) with Connes biprojectivity of A. We conclude 2010 Mathematics Subject Classification. Primary 46M10, 46H20 Secondary 46H25, 43A10,.

In this study, we show that a matrix algebra ℒ ℳ I p A is a dual Banach algebra, where A is a dual Banach algebra and 1 ≤ p ≤ 2 . We show that ℒ ℳ I p ℂ is Connes amenable if and only if I is finite, for every nonempty set I . Additionally, we prove that ℒ ℳ I p ℂ is always pseudo-Connes amenable, for 1 ≤ p ≤ 2 . Also, Connes amenability and approximate Connes biprojectivity are investigated for generalized upper triangular matrix algebras. Finally, we show that U p I p A ∗ ∗ is approximately biflat if and only if A ∗ ∗ is approximately biflat and I is a singleton.

We show that $\ell ^1(\mathbb {N}_\wedge )$ is $\varphi $ -amenable for each multiplicative linear functional $\varphi :\ell ^1(\mathbb {N}_\wedge )\rightarrow \mathbb {C}.$ This is a counterexample to the final corollary of Jaberi and Mahmoodi [‘On $\varphi $ -amenability of dual Banach algebras’, Bull. Aust. Math. Soc.105 (2022), 303–313] and shows that the final theorem in that paper is not valid.

In this paper, we introduce and study the notion of left [Formula: see text]-essential Connes amenable for dual Banach algebras. We investigate the hereditary properties of this new concept and we give some results for [Formula: see text]-Lau product and module extension. For unital dual Banach algebras, we show that left [Formula: see text]-essential Connes-amenability and left [Formula: see text]-Connes amenability are equivalent. Finally, with various examples, we examined this concept for upper triangular matrix algebras and [Formula: see text]-direct sum of 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, M I (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.