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

Let $S$ be a semitopological semigroup and $\mathcal{C}\mathcal{B}(S)$ denotes the $C^{\ast }$-algebra of all bounded complex valued continuous functions on $S$ with uniform norm. A function $f\in \mathcal{C}\mathcal{B}(S)$ is left multiplicative \linebreak continuous if and only if ${\mathbf{T}}_{\mu }f\in \mathcal{C}\mathcal{B}(S)$ for all $\mu$ in the spectrum of $\mathcal{C}\mathcal{B}(S)$, where ${\mathbf{T}}_{\mu }f(s)=\mu (L_{s}f)$ and $L_{s}f(x)=f(sx)$ for each $s,x\in S$. 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 \linebreak $(\varepsilon ,\s)$,that$ \s $isthespectrumof$Lmc(S)$,forsemitopologicalsemigroups$S$\mathrm{.}ItisshownthatifSisalocallycompactsemitopologicalsemigroup,then$S^*=\s \setminus \varepsilon(S)$isaleftidealof$\s$ if and only if for each $x,y\in S$, there exists a compact zero set $A$ such that $x\in A$ and $\{t\in S:yt\in A\}$ is a compact set.