On Amenability-Like Properties of a Class of Matrix Algebras

Abstract

In this study, we show that a matrix algebra $\mathrm{ℒ}{\mathrm{ℳ}}_I^p\left(\mathcal{A}\right)$ is a dual Banach algebra, where $\mathcal{A}$ is a dual Banach algebra and $1\le p\le 2$ . We show that $\mathrm{ℒ}{\mathrm{ℳ}}_I^p\left(ℂ\right)$ is Connes amenable if and only if $I$ is finite, for every nonempty set $I$ . Additionally, we prove that $\mathrm{ℒ}{\mathrm{ℳ}}_I^p\left(ℂ\right)$ is always pseudo-Connes amenable, for $1\le p\le 2$ . Also, Connes amenability and approximate Connes biprojectivity are investigated for generalized upper triangular matrix algebras. Finally, we show that $\mathcal{U}{\mathrm{p}}_I^p{\left(\mathcal{A}\right)}^{\ast \ast }$ is approximately biflat if and only if ${\mathcal{A}}^{\ast \ast }$ is approximately biflat and $I$ is a singleton.