On Amenability-Like Properties of a Class of Matrix Algebras

Abstract

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.