Lmc-compactification of a semitopological semigroup as a space of e-ultrafilters

Original · EN

Let S be a semitopological semigroup and CB(S) denotes the C*-algebra of all bounded complex valued continuous functions on S with uniform norm. A function f∈ CB(S) is left multiplicative continuous if and only if Tμf∈ CB(S) for all μ in the spectrum of CB(S), where Tμf(s)=μ(Lₛf) and Lₛf(x)=f(sx) for each s,x∈ S. The collection of all left multiplicative continuous functions on S is denoted by Lmc(S). In this paper, the Lmc-compactification of a semitopological semigroup S is reconstructed as a space of e-ultrafilters. This construction is applied to obtain some algebraic properties of (ε,), that is the spectrum of Lmc(S), for semitopological semigroups S. It is shown that if S is a locally compact semitopological semigroup, then S*= ε(S) is a left ideal of if and only if for each x,y∈ S, there exists a compact zero set A such that x∈ A and {t∈ S:yt∈ A} is a compact set.