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Character amenability of Banach algebras. (English) Zbl 1153.46029
The author is motivated by the notion of left amenability introduced by A. T. Lau in [Fundam. Math. 118, 161–175 (1983; Zbl 0545.46051)] for introducing the concept of character amenability as follows. A complex Banach algebra \(A\) is said to be left character amenable if for every multiplicative linear functional \(\varphi\colon A\rightarrow\mathbb{C}\) and every Banach \(A\)-bimodule \(X\) for which the right module action is given by \(x\cdot a=\varphi(a)x\) \((a\in A, \;x\in X)\), all continuous derivations from \(A\) into the dual of \(X\) are inner. Right character amenability is defined in a similar way and the Banach algebra \(A\) is called character amenable if it is both left and right character amenable.
It is shown that the character amenability of either the group algebra \(L^1(G)\) or the Fourier algebra \(A(G)\) for a locally compact group \(G\) is equivalent to the amenability of \(G\). It is also shown that the character amenability of the measure algebra \(M(G)\) is equivalent to \(G\) being a discrete amenable group. In addition, functorial properties of character amenability are studied. It should be pointed out that character amenability is closely related to the concept of amenability with respect to a character introduced and studied by E. Kaniuth, A. T. Lau and J. Pym [Math. Proc. Camb. Philos. Soc. 144, No. 1, 85–96 (2008; Zbl 1145.46027); J. Math. Anal. Appl. 344, No. 2, 942–955 (2008; Zbl 1151.46035)].

MSC:
46H05 General theory of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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