Translation invariant linear functionals on segal algebras

Abstract

Let S(G) be a Segal algebra on an infinite compact Abelian group G. We study the existence of many discontinuous translation invariant linear functionals on S(G). It is shown that if G/CG contains no finitely generated dense subgroups, then the dimension of the linear space of all translation invariant linear functionals on S(G) is greater than or equal to 2C and there exist 2C discontinuous translation invariant linear functionals on S(G), where c and CG denote the cardinal number of the continuum and the connected component of the identity in G, respectively.