On maximal nondetermining subalgebras of group algebras

Abstract

Introduction. Let L'(G) be the group algebra of an arbitrary locally compact abelian group G. It is a regular semisimple self-adjoint commutative Banach algebra with the LI-norm. The space of regular maximal ideals of L'(G) can be identified with the dual group (character group)P=6 of G. The Gelfand transform of a function f in L'(G) coincides then with the Fourier transform] Off. A subalgebra A of L'(G) is called nondetermining if A is not uniformly dense in Co(r) (cf. [2]). It is called irreducible if it does not contain any closed ideal of L'(G) with hull contained in proper closed subgroup of r. In case A does not contain any ideal of L'(G) at all, it is called completely irreducible. In this paper, we will first show that there is a reduction of maximal nondetermining subalgebras to irreducible ones. The reduced algebra will be a closed subalgebra of L'(G') where G' is a quotient group of G. Next we proved that for certain maximal nondetermining subalgebra A, we have S(A) =) c M(A). Finally, we exhibit a maximal subalgebra of L'(Z??) which is nondetermining and which does not belong to the two categories of maximal subalgebras previously known for discrete locally compact abelian groups, namely, a maximal subalgebra of L'(Z*) which is neither associated with an order of G nor gotten from an essential maximal subalgebra of C(E) for a suitable Cantor set E in r. Throughout this paper, we adopt the terminology and notations of Rudin [4] and Hoffman & Singer [3].