Embedding Locally Compact Semigroups into Groups

Abstract

Let X, Y, Z be topological spaces. A function F :X × Y → Z is called jointly continuous if it is continuous from X × Y with the product topology to Z . It is said to be separately continuous if x 7→ F (x, y):X → Z is continuous for each y ∈ Y and y 7→ F (x, y):Y → Z is continuous for each x ∈ X . A semitopological semigroup is a semigroup S endowed with a topology such that the multiplication function is separately continuous, or equivalently, all left and right translations are continuous. In the case the semigroup is actually a group, we call it a semitopological group (it is often required that the inversion function be continuous also, but we omit this assumption). There is considerable literature (see [5] for a survey and bibliography) devoted to the problem of embedding a topological semigroup (a semigroup with jointly continuous multiplication) into a topological group, but the semitopological version appears to have received scant attention. On the other hand there are good reasons for considering this case, among them the fact that a number of general conditions exist for concluding that a semitopological group is actually a topological one. We further find that the semitopological setting gives much more straightforward statements and proofs of key results. The ideas of this paper parallel in many aspects those contained in the first part of [1]. Our Corollary 2.7 is essentially Theorem 2.1 of [1] extended from right reversible semigroups to general semigroups embedded in a group. Results paralleling most of the results of this paper can be found in Chapter VII of [4], except that we relax the hypothesis assumed on the semigroup S to only assuming translations are open mappings (a stronger condition on the semigroups S is assumed in [4] to guarantee continuity of inversion in the containing group). In addition, our methods here are much quicker and more direct. The algebraic problem of giving necessary and sufficient conditions for group embeddability of a semigroup is a delicate one, although cancellativity is an obvious necessary condition. We bypass the algebraic problem by considering only semigroups that are (algebraically) group embeddable. If a semigroup S embeds in a group G , then there is a smallest subgroup in G containing S and we assume that our embeddings are always readjusted at the codomain level so that S generates (as a group) G .