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Inner amenable locally compact groups. (English) Zbl 0718.43002

The locally compact group G is defined to be amenable [resp. inner amenable] if there exists a mean M on \(L^{\infty}(G)\) such that for all \(a\in G\), \(f\in L^{\infty}(G)\) one has \(M(_ af)=M(f)\) [resp. \(M(_ af_{a^{-1}})=M(f)]\), where \({}_ af_ b(g)=f(agb)\), a,b,g\(\in G\). The von Neumann algebra \({\mathcal M}\) of operators on a Hilbert space is said to have property (P) if for every \(T\in {\mathcal M}\), the commutant of \({\mathcal M}\) intersects nonvacuously the weak \({}^*\)-closed convex hull of \(U^*TU\), where U runs over the set of unitary operators in \({\mathcal M}.\)
Let now G act invertibly on a locally compact Hausdorff space X and consider the von Neumann algebra \({\mathcal L}\) generated by operators \(V_ a\), \(N_{\Phi}\) (a\(\in G\), \(\Phi \in L^{\infty}(G))\) on the Hilbert space \(L^ 2(X\times G)\), X being endowed with a quasi-invariant nonnegative Radon measure; \(V_ af(x,b)=f(x,a^{-1}b)\), \(N_{\Phi}f(x,b)=\Phi (xb^{-1})f(x,b)\) with \(f\in L^ 1(X\times G)\), \(x\in X\), \(b\in G\). The authors show that G is amenable if and only if G is inner amenable, \(L^{\infty}(X)\) admits a G-invariant mean and \({\mathcal L}\) is injective, i.e., admits property (P). More results are established linking the concepts of amenability, inner amenability, property (P) and providing insight into the phenomenon of inner amenability.
Reviewer: J.-P.Pier (Esch)

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
46L10 General theory of von Neumann algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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