  In this paper we introduce a new notion of strong pseudo-amenability for
Banach algebras. We study strong pseudo-amenability of some Matrix algebras.
Using this tool, we characterize strong pseudo-amenability of $\ell^{1}(S)$,
provided that $S$ is a uniformly locally finite semigroup. As an application we
show that for a Brandt semigroup $S=M^{0}(G,I)$, $\ell^{1}(S)$ is strong
pseudo-amenable if and only if $G$ is amenable and $I$ is finite. We give some
examples to show the differences of strong pseudo-amenability and other
classical notions of amenability.
