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Invariant subspaces for algebras of linear operators and amenable locally compact groups. (English) Zbl 0645.43002

Let G be a locally compact group and denote by \(L^ 1(G)\), M(G), A(G) and B(G) the group algebra, the measure algebra, the Fourier algebra and the Fourier-Stieltjes algebra of G, respectively. All these Banach *- algebras share a common property: each of them is the predual of a \(W^*\)-algebra, and the identity of the \(W^*\)-algebra is a multiplicative linear functional on A. This fact motivated the first author to define a new class of Banach algebras, so-called F-algebras, by the abstract property just mentioned [Fundam. Math. 118, 161-175 (1983; Zbl 0545.46051)]. Now let A be an F-algebra, and let \(S_ A\) denote the set of all positive functionals in \(A\subseteq A^{**}\) with norm one. A is called left amenable if \(A^*\) has a topological left invariant mean, i.e. a positive linear functional of norm one such that \(m(F\cdot a)=m(F)\) for \(F\in A^*\) and \(a\in S_ A\), where \(F\cdot a\in A^*\) is defined by \(<F\cdot a,b>=<F,ab>\) for \(b\in A.\)
The purpose of the paper under review is to characterize left amenability of A in terms of a certain invariant subspace property T(n) for n- dimensional subspaces of separated locally convex representation spaces of A. A similar invariant subspace property for semigroups has previously been studied by K. Fan [Indagationes Math. 27, 447-451 (1965; Zbl 0139.311)] and A. T. M. Lau [J. Math. Anal. Appl. 97, 374-379 (1983; Zbl 0549.43003)]. As consequences of their main theorem the authors obtain the following results for locally compact groups G. A(G) and B(G) satisfy property T(n), and the same is true for \(L^ 1(G)\) and M(G) provided that G is amenable. Conversely, if T(1) holds for either \(L^ 1(G)\) or M(G), then G is amenable.
Reviewer: E.Kaniuth

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
43A10 Measure algebras on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
47A15 Invariant subspaces of linear operators
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