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  • Author or Editor: Hojatollah Samea x
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In the present paper we have introduced Banach algebras \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\ell ^p - \oplus _{i \in I} \mathfrak{A}_i $$ \end{document} (1 ≦ p < ∞) of unital Banach algebras \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathfrak{A}_i ,\mathcal{M}_\infty ^0 (\mathfrak{A},I)$$ \end{document}, and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{M}^0 (\mathfrak{A},I)$$ \end{document}, in order to investigate the amenability and essential amenability of the Banach algebras \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathfrak{E}_p (I)$$ \end{document} (1 ≦ p < ∞), the convolution Banach algebra A ( G ) of a compact group G , and the ℓ 1 -Munn algebra \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{L}\mathcal{M}_I (\mathfrak{A})$$ \end{document}. Examples are given to establish negatively parts of the open problems raised by Ghahramani and Loy concerning the essential amenability of Banach algebras. Examples of semigroups S are given to prove that the amenability of S is neither sufficient nor necessary for the essential amenability of the semigroup algebra ℓ 1 ( S ).

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In this paper we introduce the notions of essentially left amenable and approximately left amenable Lau algebras together with several characterizations of such algebras. In particular we investigate the relations between the left amenability, the essential left amenabil-ity and the approximate left amenability of certain quotient Lau algebras as well as operator projective tensor product of Lau algebras. The obtained results are applied to prove that the three notions coincide for a number of well-known semigroup algebras.

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