Weak and Approximate Amenability of Second Dual of Foundation Semigroup Algebras

Document Type : Original Article

Author

University of Tehran, Tehran, Iran

10.22091/maa.2026.17170.1066

Abstract

In this paper, we study the weak and approximate amenability of the second dual of foundation semigroup algebras endowed with the first Arens product. Let $\mathcal S$ be a compactly cancellative foundation topological semigroup with identity element $e$. We establish that the weak or approximate amenability of $LUC(\mathcal S)^{\ast}$ or $(M_a(\mathcal S)^{\ast \ast}, \Box)$ implies the corresponding amenability property for the measure algebra $M(\mathcal S)$. Furthermore, we show that if the second dual algebra $(M_a(\mathcal S)^{\ast \ast}, \Box)$ is approximately amenable, then $\mathcal S$ must be discrete and left amenable. Finally, for a cancellative, compactly cancellative foundation topological semigroup with identity, we prove that the approximate amenability of $(M_a(\mathcal S)^{\ast \ast}, \Box)$, the amenability of

$(M_a(\mathcal S)^{\ast \ast}, \Box)$, and the finiteness of $\mathcal S$ are all equivalent.

Keywords

Main Subjects

Measure Algebras and Applications

Articles in Press, Accepted Manuscript
Available Online from 13 October 2026

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