Arens regularity and discrete groups.

*(English)*Zbl 0746.43002The 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 \(G\).

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 \(G\).

Reviewer: J.S.Pym (Sheffield)

##### 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 |