Linear representations of certain compact semigroups

Abstract

Abstract. In this paper we initiate the study of representation theory of compact, not necessarily commutative, uniquely divisible semigroups. We show that a certain class of semigroups are all topologically isomorphic to real matrix semigroups. The proof utilizes a group embedding theorem and the standard results on homomorphisms of Lie groups into matrix groups.

Embedding Locally Compact Semigroups into Groups

“…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.…”

“…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).…”

“…Then by passing to a small enough neighborhood, one finds a euclidean neighborhood U of x with compact closure. One can use the theorem of Invariance of Domain to establish that U is translatable (see, for example, the argument in the proof of Theorem 2.1 of [1]). One concludes that S • ⊆ I 0 .…”

Product semigroups

“…Hence there exist y 6 G 5 2 and z 6 E S 3 such that xyzt = uy 6 Brown and Friedberg (1971) if W is a semigroup defined on a closed subset of Euclidean n-space E" having non-empty interior in E" and is left (right) reversible, cancellative, and translation functions are homeomorphisms into, then W is embedded in a Lie group. The semigroup T satisfies this, thus is embedded in a Lie group.…”