Compact left ideal groups in semigroup compactification of locally compact groups

Abstract

LetGbe a locally compact group and letUGdenote the spectrum of theC*-algebraLUC(G)of bounded left uniformly continuous complex-valued functions onG, with the Gelfand topology. Then there is a multiplication onUGextending the multiplication onG(when naturally embedded inUG) such thatUGis a semigroup and for eachx ∈ UG, the mapy ↦ yxfromUGintoUGis continuous, i.e.UGis a compact right topological semigroup. ConsequentlyUGhas a unique minimal idealKwhich is the union of minimal (closed) left idealsUG. FurthermoreKis the union of the set of maximal subgroups ofK(see [3], theorem 3·ll).