Bohr compactification

Abstract: Let G and H be locally compact abelian groups with character groups G*, H*, and let denote the pairing between a group and its dual. In 1952 Kaplansky proved the following result, using the structure of locally compact abelian groups and category arguments. Theorem 1.1. Let τ: G → H be an algebraic homomorphism for which there is a dual τ* : H* → G* (so that = for all g in G, h* in H*). Then τ is continuous. The result bears a striking similarity to a well-known fact about Banach spaces which is a consequence of uniform boundedness; the present note is devoted to an analogous “uniform boundedness” for groups, which yields a non-structural proof of Kaplansky's theorem.

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.

Abstract: The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg (G): If G is a locally compact Abelian group with Bohr compact- ification bG, and if N is a closed metrizable subgroup of bG, then every A G satisfies: A · (N \ G) is compact in G if and only if {aN : a 2 A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N \ G = {1}; (b) the asserted equivalence can hold for suitable G and N with N closed in bG but not metrizable; (c) an Abelian group may admit two topological group topologies U and T , with U totally bounded, T locally compact, U T , with U and T sharing the same compact sets, and such that nevertheless U is not the topology inherited from the Bohr compactification of hG,T i. There are applications to topological groups of the form kG for G a totally bounded Abelian group.