Amenable Banach Algebras
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We now move from groups to Banach algebras. There are various Banach spaces and algebras associated with a locally compact group G: see Appendices D and F. Perhaps the most important one is the group algebra \(L^1(G)\). It is a complete invariant for G: if \(G_1\) and \(G_2\) are locally compact groups such that \(L^1(G_1)\) and \(L^1(G_2)\) are isometrically isomorphic, then \(G_1\) and \(G_2\) are topologically isomorphic, i.e., all information about G is already encoded in \(L^1(G)\).