Pseudo-amenability and Biflatness of weighted algebras and their second dual on inverse semigroups

Document Type : Original Article

Author
Kobra Oustad
Department of mathematics, Dehdasht Branch, Islamic Azad University, Dehdasht, Iran. P. O. Box 7571763111, Dehdash, Iran.
10.22034/maco.5.1.7
Abstract
This paper proves that when $S$ is an inverse semigroup, pseudo- menability, amenability and approximately amenability of $\ell^{1}(S, \omega)$ are equivalent, $E(S)$ is finite and every maximal subgroup of $S$ is amenable.
In addition, for a discrete inverse semigroup of $S$, we prove when $(E(S),\leq)$ is niformly locally finite, then $\ell^{1}(S, \omega)^{**}$ is pseudo-amenable if and only if $\ell^{1}(S)$ , it is pseudo-amenable and $S$ is finite.
Also , we prove that for discrete inverse semigroup $S$, if $\ell^{1}(S, \omega)$ has a bounded approximate identity then $\ell^{1}(S, \omega)^{**}$ is amenable if and only if $\ell^{1}(S)$ is biprojective and $S$ is finite. Also, we show that for an Archimedean semigroup $S$, if $\Omega$ be bounded on every maximal subgroup $G$ of $S$, pseudo-amenability,amenability and approximate amenability of $\ell^{1}(S, \omega)$ are equivalent, and also if $S$ be a weakly cancellative commutative semigroup, we obtain some results on pseudo-amenability of the second dual of $\ell^{1}(S, \omega)$ .

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