We investigate -approximate contractibility and -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where is a dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.
In this paper, we define the notion of sigmaapproximate module amenability of Banach algebras and give some properties about this notion. Also for Banach Abimodule A, and J , the closed ideal of A generated by elements of form ða Á aÞb À aðb Á aÞ, ða; b 2 A; a 2 AÞ, we considered some corollaries about b r-approximate amenability of A J as a Banach A-bimodule, where b r : A J ! A J by b rða þ J Þ ¼ rðaÞ þ J has a dense range.
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