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Some notes on \((\sigma,\tau)\)-amenability of Banach algebras. (English) Zbl 1199.46111
Let \(A\) be a Banach algebra and let \(\sigma\), \(\tau\) be continuous homomorphisms on \(A\). The authors define a \((\sigma,\tau)\)-derivation from \(A\) into a Banach \(A\)-bimodule \(X\) to be a linear mapping \(d\colon A\to X\) satisfying the identity \(d(ab)=d(a)\sigma(b)+\tau(a)d(b)\) for all \(a,b\in A\). Such \(d\) is said to be inner if it is implemented by an element \(x\in X\) in the following sense: \(d(a)=x\sigma(a)-\tau(a)x\) for all \(a\in A\). The authors study the concept of \((\sigma,\tau)\)-amenability of \(A\) (every such \(d\) into a dual module is inner) including notions of \(\sigma\)-virtual diagonal and \(\sigma\)-approximate diagonal.

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
47B47 Commutators, derivations, elementary operators, etc.