Georgian Mathematical Journal, ( ISI ), Volume (18), No (1), Year (2011-5) , Pages (137-145)

Title : ( σ,τ-amenability of C*-algebras )

Authors: Madjid Mirzavaziri , Mohammad Sal Moslehian ,

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Abstract

Suppose that ${\\mathcal A}$ is an algebra, $\\sigma,\\tau:{\\mathcal A}\\to{\\mathcal A}$ are two linear mappings such that both $\\sigma({\\mathcal A})$ and $\\tau({\\mathcal A})$ are subalgebras of ${\\mathcal A}$ and ${\\mathcal X}$ is a $\\big(\\tau({\\mathcal A}),\\sigma({\\mathcal A})\\big)$-bimodule. A linear mapping $D:{\\mathcal A}\\to {\\mathcal X}$ is called a $(\\sigma,\\tau)$-derivation if $D(ab)=D(a)\\cdot\\sigma(b)+\\tau(a)\\cdot D(b)\\\\\\,(a,b\\in {\\mathcal A})$. A $(\\sigma,\\tau)$-derivation $D$ is called a $(\\sigma,\\tau)$-inner derivation if there exists an $x\\in{\\mathcal X}$ such that $D$ is of the form either $D_x^-(a)=x\\cdot \\sigma(a)-\\tau(a)\\cdot x\\,\\,(a\\in {\\mathcal A})$ or $D_x^ +(a)=x\\cdot \\sigma(a)+\\tau(a)\\cdot x \\,\\,(a\\in {\\mathcal A})$. A Banach algebra ${\\mathcal A}$ is called $(\\sigma,\\tau)$-amenable if every $(\\sigma,\\tau)$-derivation from ${\\mathcal A}$ into a dual Banach $\\big(\\tau({\\mathcal A}),\\sigma({\\mathcal A})\\big)$-bimodule is $(\\sigma,\\tau)$-inner.\\\\ Studying some general algebraic aspects of $(\\sigma,\\tau)$-derivations, we investigate the relation between amenability and $(\\sigma,\\tau)$-amenability of Banach algebras in the case when $\\sigma, \\tau$ are homomorphisms. We prove that if $\\mathfrak A$ is a $C^*$-algebra and $\\sigma, \\tau$ are $*$-homomorphisms with $\\ker(\\sigma)=\\ker(\\tau)$, then ${\\mathfrak A}$ is $(\\sigma, \\tau)$-amenable if and only if $\\sigma({\\mathfrak A})$ is amenable.

Keywords

, , τ)-amenability of C*-algebras
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@article{paperid:1022035,
author = {Mirzavaziri, Madjid and Sal Moslehian, Mohammad},
title = {σ,τ-amenability of C*-algebras},
journal = {Georgian Mathematical Journal},
year = {2011},
volume = {18},
number = {1},
month = {May},
issn = {1072-947X},
pages = {137--145},
numpages = {8},
keywords = {(σ;τ)-amenability of C*-algebras},
}

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%0 Journal Article
%T σ,τ-amenability of C*-algebras
%A Mirzavaziri, Madjid
%A Sal Moslehian, Mohammad
%J Georgian Mathematical Journal
%@ 1072-947X
%D 2011

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