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

Let S be a semitopological semigroup and CB(S) denote 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 the 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, S), such that S is the spectrum of Lmc(S), for semitopological semigroups S. It is shown that if S is a locally compact semitopological semigroup, then S∗ = S \ ε(S) is a left ideal of S if and only if for each x, y ∈ S, there exists a compact zero set A containing x such that {t ∈ S : yt ∈ A} is a compact set.