Projections and approximate identities for ideals in group algebras

Abstract

For a locally compact group G with property (PI), if there is a continuous projection of L1(G) onto a closed left ideal I, then there is a bounded right approximate identity in I. If I is further 2-sided, then I has a 2sided approximate identity. The converse is proved for w*-closed left ideals. Let G be further abelian and let I be a closed ideal in L 1(G). The condition that I has a bounded approximate identity is characterized in a number of ways which include (1) the factorability of I, (2) that the hull of I is in the discrete coset ring of the dual group, and (3) that I is the kernel of a closed element in the discrete coset ring of the dual group. Introduction. Let G be a locally compact group, I a closed left ideal in L l(G) and P a continuous projection of L 1(G) onto I. It is proved by W. Rudin [I1, Theorem 1] that, if G is compact, there exists a continuous projection Q of L1(G) onto I such that (*) / * Qg = Q (/ * g) (/, g E L1(G)). Further [1, Proof of Theorem 2], if in addition G is abelian, then there exists an idempotent measure p on G such that Qf = f* t (f cL (G)) so that Q is actually an algebra homomorphism. It follows that I, considered as a Banach algebra, has a bounded approximate identity. The purpose of Part I of this paper is to find out what happens if G is not compact or abelian. It turns out that if G has the property (P ) (which it does if it is compact) then the projection P leads to a net of projections Q for which the formula (*) almost holds, and that I still has a bounded (right) approximate identity (Theorem 2). Received by the editors December 10, 1971. AMS (MOS) subject classifications (1970). Primary 22B10, 22D15, 43A20; Secondary 43A45, 46H 10.