On Amenability-Like Properties of a Class of Matrix Algebras

Abstract

In this study, we show that a matrix algebra ℒℳIpA is a dual Banach algebra, where A is a dual Banach algebra and 1≤p≤2. We show that ℒℳIpℂ is Connes amenable if and only if I is finite, for every nonempty set I. Additionally, we prove that ℒℳIpℂ 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 UpIpA∗∗ is approximately biflat if and only if A∗∗ is approximately biflat and I is a singleton.