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Amenability with respect to a closed subgroup of a product group. (English) Zbl 0763.43002

The concept of \(\Gamma\)-amenability of locally compact groups is introduced, which is a generalization of the left amenability and the linear amenability in the usual sense. Let \(G\) be a locally compact group with unit element \(e\), and \(\Gamma\) be a closed subgroup of the product group \(G\times G\). The group \(G\) is said to be \(\Gamma\)-amenable if there exists a mean \(M\) on \(L^ \infty(G)\) such that \(M({_ af_ b})=M(f)\) for any \(f\in L^ \infty(G)\) and \((a,b)\in\Gamma\), where \({_ af_ b}(x)=f(a^{-1}xb)\) (\(x\in G\)). In the case \(\Gamma=G\times\{e\}\) and \(\Gamma=\{(x,x)\); \(x\in G\}\), \(\Gamma\)-amenability is equivalent to the left amenability and the inner amenability respectively. Various characterizations of \(\Gamma\)-amenability are established, which are analogous to the characterizations of amenability.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
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