In this paper, we introduce a notion of ultra central approximate identity for Banach algebras which is a generalization of the bounded approximate identity and the central approximate identity. Using this concept we study pseudo-contractibility of some matrix algebras among $ell^1$-Munn algebras. As an application, for the Brandt semigroup $S=M^{0}(G,I)$ over a non-empty set $I$, we show that $ell^{1}(S)$ has an ultra central approximate identity if and only if $I$ is finite. Also we show that the notion of pseudo-contractibility and contractibility are the same on $ell^{1}(S)^{**}$, where $S$ is the Brandt semigroup.

Amir Sahami

Mehdi Rostami

On pseudo-contractibility of certain algebras related to a discrete semigroup

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