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Group actions whose space of invariant means is finite dimensional. (English) Zbl 1429.43002

A bilinear map between Banach spaces extends to a bilinear map on the second duals in two different ways, as it was introduced by R. Arens for the first time [Proc. Am. Math. Soc. 2, 839–848 (1951; Zbl 0044.32601)]. Since then, measuring the difference of these two extensions has been a major question, with emphasis on two particular types of bilinear maps, namely Banach algebra products and module actions. Let us call the two extensions of bilinear maps, the first and second Arens products. The paper under review deals with certain module actions and their Arens extensions. Let \(G\) be a group which acts on a set \(X\). This action turns \(\ell^1(X)\) into a left \(\ell^1(G)\) module in a natural way. A positive element \(n\) of \(\ell^\infty(X)^*\) such that \(n(1)=1\) and \(g\odot_1 n=n\) for every \(g\in G\) where \(\odot\) denotes the first Arens product, is called a \(G\)-invariant mean on \(\ell^\infty(X)\). The action of \(G\) on \(X\) is called amenable if there is a \(G\)-invariant mean on \(\ell^\infty(X)\) and is called uniquely amenable if there exists a unique \(G\)-invariant mean on \(\ell^\infty(X)\).
The paper under review investigates the existence and uniqueness of \(G\)-invariant means on the set \(\mathbb{N}\) of natural numbers, when \(G\) is a subgroup of the permutation group \(S_\infty\) on \(\mathbb{N}\). More precisely, it is shown that under certain set theoretic assumptions, there is a locally finite subgroup \(G\) of \(S_\infty\) which has a unique \(G\)-invariant mean. Then it is shown that non-uniqueness of \(G\)-invariant means is also possible, by showing that under certain assumptions there is a locally finite subgroup \(G\) of \(S_\infty\) for which there are two distinct \(G\)-invariant means on \(\ell^\infty(X)\). The difference of two Arens products is reflected in the weak*-weak* continuity of left and right translations. Let \(\mathcal{A}\) be a Banach algebra, \(\mathcal{B}\) be a left Banach \(\mathcal{A}\)-module and \(\odot_1\) and \(\odot_2\) denote the first and second Arens products derived from the module action of \(\mathcal{A}\) on \(\mathcal{B}\), respectively. The first and second topological centers of the module action are the sets \begin{align*} \Lambda_1&=\{ m\in\mathcal{A}^{**}\mid \text{the mapping } n\to m\odot_1 n \text{ is continuous from } \mathcal{B}^{**} \text{ to itself}\} \\ \Lambda_2&=\{ n\in\mathcal{B}^{**}\mid \text{ the mapping }m\to m\odot_1 n\text{ is continuous from } \mathcal{A}^{**}\text{ to }\mathcal{B}^{**}\}. \end{align*} The authors find an amenable subgroup \(F\) of \(S_\infty\) such that the set of \(F\)-invariant means on \(\ell^\infty(\mathbb{N})\) is finite dimensional, but the set of such means does not intersect \(\Lambda_1\). The paper is concluded with some remarks and open problems.

MSC:

43A10 Measure algebras on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.

Citations:

Zbl 0044.32601
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