Finite-dimensional invariant subspace property and amenability for a class of Banach algebras (Q2790607)

This interesting paper extends \textit{K. Fan}'s remarkable invariant subspace theorem for left amenable semigroups in [Bull. Am. Math. Soc. 69, 773--777 (1963; Zbl 0118.10803)] to a similar result for the class of \(F\)-algebras.NEWLINENEWLINE\(F\)-algebras, also known as Lau algebras, introduced by the first named author in [Fundam. Math. 118, 161--175 (1983; Zbl 0545.46051)], are those Banach algebras \(A\) whose dual is a von Neumann algebra with the identity \(e\) a character of \(A\). This class includes the group algebra \(L^1(G)\), the Fourier algebra \(A(G)\) of a locally compact group \(G\), but also many other classes of algebras that are studied in abstract harmonic analysis. In that paper, the first named author also defined the notion of left-amenable for an \(F\)-algebra \(A\) as the condition that every bounded derivation \(A\to X^*\) is inner whenever \(X\) is a Banach \(A\)-bimodule whose left multiplication is given by \(\varphi \cdot x=\varphi(e)x\) for all \(\varphi\in A\) and \(x\in X\).NEWLINENEWLINEThe main result of the paper under review is the following theorem. But before stating this theorem, we need an additional terminology: Let \(E\) be a locally convex vector space, and let \({\mathcal S}=\{T_s: s\in S\}\) be a linear representation of a semigroup \(S\) on \(E\). A subset \(X\) of \(E\) is \(n\)-consistent with respect to \({\mathcal S}\) for some \(n\in\mathbb N\) if \(X\) contains an \(n\)-dimensional subspace of \(X\) and every \(T_s\) maps every \(n\)-dimensional subspace of \(X\) to another one.NEWLINENEWLINETheorem. Let \(A\) be an \(F\)-algebra, and consider the semigroup \(S\) of all the normal states of \(A^*\). Then the left-amenability of \(A\) is equivalent to each of the following \(n\)-dimensional invariant subspace properties \((F_n)\): Let \({\mathcal S}=\{s\mapsto T_s\}\) be a linear representation of \(S\) on a locally convex space \(E\) that is jointly continuous on compact subsets of \(E\) such that the mapping \(s\mapsto T_s(x)\) is continuous for each fixed \(x\in E\). If \(X\) is a subset of \(E\) that is \(n\)-consistent with respect to \({\mathcal S}\) and if there exists a closed \({\mathcal S}\)-invariant subspace \(H\) in \(E\) of codimension \(n\) with the property that \((x+H)\cap X\) is compact for each \(x\in E\), then there exists an \(n\)-dimensional subspace \(L_0\) contained in \(X\) such that \(T_s(L_0)=L_0\) for all \(s\in S\).NEWLINENEWLINEOn the way to prove this theorem, the authors also prove that an \(F\)-algebra \(A\) is left amenable if and only if the semigroup \(S\) of all the normal states of \(A^*\) is extremely left amenable in the sense that the \(C^*\)-algebra \(LUC(S)\) of all left uniformly continuous functions on \(S\) possesses a left invariant mean that is multiplicative.