Closed ideals in the group algebra 𝐿¹(𝐺)∩𝐿²(𝐺)

Abstract

0. Introduction. In the following, G will denote a locally compact abelian topological group with character group For 1 < p < oo, LP(G) is the Banach space of all complex-valued functions whose pth powers are Haar integrable over (LP(G) is often written LP when the group G is obvious from the context.) The linear space L'(G) n L2(G) (denoted LI n L2) is normed in such away that, under convolution as multiplication, it is a commutative Banach algebra (?2). It is also proved in ?2 that it is regular, semi-simple and that its regular maximal ideal space is 6. Itis shown (?3) that the abstract Silov theorem [8, p. 86] holds for LI n L2. The standard proof of this theorem in L'(G) seems to depend upon the uniform boundedness of the approximate identity. A novel aspect of the LI nl L2 case is that a similar proof is obtained despite the fact that every approximate identity in L' rn L2 is unbounded. An important but unsolved problem of harmonic analysis is the classification of the closed ideals in L'(G). Using the additional structure supplied by LI n L2 it is to be expected that more precise results can be obtained about the closed ideals in LI n L2. If G and G are both locally compact metric abelian groups, examples of the more precise results that can be obtained are: (a) If I is a closed proper ideal in LI n L2, then there exists an x E I such that the hull of x and the hull of I coincide except for a set of measure zero (Theorem 7.2). (b) For every closed invariant proper subspace N c L2(G), N n L' = k(h(N n LI)) (Corollary 2 of Theorem 7.4). This permits a new characterization of the kernel of E for a class of perfect sets E c G. (A. Denjoy terms these sets epais en lui-meme in Lepons sur le calcul des coefficients d'une serie trigonometrique, Paris, 1941, 2ieme Partie, p. 100.) (c) The set f of all closed proper ideals in LI n L2 which are