Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, P.O.Box 53751-71379, Tabriz, Iran.

Abstract

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $\ell^1(S)$ is  a symmetric $\ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigroup $E(S)$ of idempotents. In symmetric connes amenability, we have proved that $M(G)$ is symmetric connes amenable if and only if $G$ is amenable.

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###### ##### References
[1] M. Amini, Module Amenability for Semigroup Algebras, Semigroup Forum, 69 (2004), 243-254.
[2] J.F. Berglund, H.D. Junghenn, and P. Milnes, Analysis on Semigroups, Wiley–Interscience, New York, 1989.
[3] H.G. Dales, F. Ghahramani, and A.Ya. Helemskii, The amenability of measure algebras, J. London Math. Soc., 66 (2001),  213-226.
[4] B.E. Johnson, Cohomology in Banach algebras, Mem.  Amer. Math. Soc., 127 (1972).
[5] B.E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Camb. Phil. Soc., 120 (1996), 455-473.
[6] V. Runde, Connes-amenability and normal, virtual diagonals for measure algebras I,  Bull. London Math. Soc. 47 (2015),  555-564.
[7]  V. Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand. 95 (2004), 124 -144.