Semigroups and de Leeuw’s convexity theorem

Abstract

In this article the convexity theorem of de Leeuw [2, p. 194] is derived by a systematic application of the theory of Banach algebras; the variables aspect is thereby considerably reduced. The proof can be even further simplified in the case of compact Reinhardt domains in Cn. de Leeuw's theorem may be stated as follows: G is an abelian group with a finite set of generators {Xl, x2, * , Xs }, F the dual group of G, and H the semigroup generated by the set given, not necessarily including the unit of H. P(H) is the semigroup algebra of H over the complex numbers. T is a compact family of homomorphisms of H into the multiplicative semigroup of complex numbers, or, as we shall say, representations of H. Define N(h) max |+(h) 6 Ei T}, 11 = 1 cihil --,= i I ci I N(hi). The norm may be indefinite or incomplete, but one can consider bounded homomorphisms of P(H) in this norm. Every complex homomorphism of P(H) is given by a representation of H, schematically, (o, >'i= cihi) = E' 1 c,a(h,). The homomorphism is continuous for the given norm exactly when ? <N. Finally, 1FT is the compact family of representations h-'y(h)o(h) with yCP, OCT. Then: THEOREM. Fr T contains the Silov boundary of (the bounded homomorphisms of) P(H).