Amenability for twisted covariance algebras and group \(C^*\)-algebras
Three notions of amenability have turned out to be important: Amenable groups, amenable Banach algebras and more specifically amenable \(C^*\)- algebras, amenable von Neumann algebras. For \(C^*\)-algebras, amenability means nuclearity; for von Neumann algebras, amenability signifies injectivity. The locally compact group \(G\) is amenable if and only if its group algebra \(L_ 1(G)\) is an amenable Banach algebra; the \(C^*\)-algebra \(A\) is amenable if and only if its second dual space \(A''\) constitutes an amenable von Neumann algebra. Now a locally compact group \(G\) is called \(C^*\)-amenable group in case its \(C^*\)-algebra \(C^*(G)\), i.e., the completion of \(L_ 1(G)\), is amenable. Every amenable group is \(C^*\)-amenable; the converse does not hold. The article describes profound investigations on stability properties inside the class of \(C^*\)-amenable groups. They are carried out in the large framework of twisted covariance algebras [\textit{P. Green}, Acta Math. 140, 191-250 (1978; Zbl 0407.46053)]. The authors take care to produce a rather self-contained and original approach to the theories of crossed products and twisted crossed products. Let \(G\) be a locally compact group and \(A\) a \(C^*\)-algebra. One considers the Banach \(^*\)-algebra \(L_ 1(G,A) = L_ 1(G)\overset\widehat{}{\smash\otimes} A\) of Bochner integrable functions from \(G\) into \(A\). Convolution of \(L_ 1 (G,A)\) is defined via a continuous group homomorphism \(\alpha : G \rightarrow Aut A\); \((G,A,\alpha)\) is a covariant system. If \(N\) is a closed normal subgroup of \(G\), one assumes the existence of a continuous homomorphism \({\mathcal T}\) of \(N\) into the unitary group \(U({\mathcal M}(A))\) of the algebra \({\mathcal M}(A)\) of multipliers on \(A\). Then \((G,A,\alpha,N,{\mathcal T})\) forms a twisted covariant system if for all \(n \in N\), \(x \in G\), \(a \in A\), \({\mathcal T}(n)a{\mathcal T}(n^{-1}) = \alpha_ n (a)\), \({\mathcal T}(xnx^{-1}) = a_ x ({\mathcal T}(n))\). Let \(C^*(G,A,{\mathcal T}) = C^*(G,A)/I\), were \(I\) is the closed ideal of \(C^*(G,A)\) generated by the elements \(f(n - {\mathcal T}(n))g\), where \(f, g \in L_ 1 (G,A)\). The authors prove that if \(H\) is a closed normal subgroup of \(G\) containing \(N\), then \(C^*(H,A,{\mathcal T})\) is amenable if \(C^*(G,A,{\mathcal T})\) is. As a corollary they deduce that if \(H\) is any closed normal subgroup of \(G\) that is \(C^*\)-amenable, so is \(G\). Also if \(H\) is a closed \(C^*\)- amenable subgroup of \(G\) and \(G/H\) is amenable, then \(G\) is \(C^*\)- amenable. In a final section, the authors term the locally compact group \(G\) projective amenable in case the von Neumann algebra \(A_ \pi(G)\) is amenable for every projective (unitary) representation \(\pi\) of \(G\) on a Hilbert space. They prove that each almost connected group is projective amenable.
scientific article; zbMATH DE number 21943
scientific article