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Arens regularity and discrete groups. (English) Zbl 0746.43002
The Herz algebra $$A_ p(G)$$ of a locally compact group $$G$$ for $$1<p<\infty$$ is the space of functions $$u$$ which can be written in the form $$u(x)=\sum_{i=1}^ n\int_ G f_ i(y)g_ i(xy)dy$$ (the integral being with respect to left Haar measure) with $$f_ i\in L_ p(G)$$, $$g_ i\in L_ q(G)$$ ($$q$$ being the index conjugate to $$p$$), with the norm of $$u$$ taken to be the infimum of all the sums $$\sum_{i=1}^ n \| f_ i\|_ p\| g_ i\|_ q$$ arising from ways of expressing $$u$$ as above. The paper is an essay on a variety of relationships between the Arens regularity of Banach algebras associated with $$A_ p(G)$$ or more particularly $$A_ 2(G)$$, the existence of invariant means on various spaces, and properties of the dual of $$A_ p(G)$$ and of the group $$G$$. Here, we provide only a sample of the conclusions presented.
The first result is that the Arens regularity of any $$A_ p(G)$$ implies that $$G$$ is discrete; this corrects a claim by the reviewer and E. Oshobi [Math. Proc. Camb. Philos. Soc. 102, 481-505 (1987; Zbl 0663.46045)]. If $$G$$ is amenable and $$A_ p(G)$$ is Arens regular, then $$G$$ is finite if and only if $$A_ p(G)$$ is weakly sequentially complete (so that in particular if $$A_ 2(G)$$ is Arens regular, $$G$$ must be finite, a result due to A. Lau and J. Wong [Proc. Am. Math. Soc. 107, 1031-1036 (1989; Zbl 0696.43001)]). The author provides a long list of other conditions on $$G$$ under which Arens regularity of $$A_ 2(G)$$ is equivalent to the finiteness of $$G$$.

##### MSC:
 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 43A07 Means on groups, semigroups, etc.; amenable groups 43A40 Character groups and dual objects 43A75 Harmonic analysis on specific compact groups
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