Trivial extensions of semigroups and semigroup $C^*$-algebras

The object of the study in the paper is reduced semigroup $C^*$-algebras for left cancellative semigroups. Such algebras are a very natural object because it is generated by isometric shift operators belonging to the image of the left regular representation of a left cancellative semigroup. These operators act in the Hilbert space consisting of all square summable complex-valued functions defined on a semigroup. We study the question on functoriality of involutive homomorphisms of semigroup $C^*$-algebras, that is, the existence of the canonical embedding of semigroup $C^*$-algebras induced by an embedding of corresponding semigroups. In order to do this, we investigate the reduced semigroup $C^*$-algebras associated with semigroups involved in constructing normal extensions of semigroups by groups. At the same time, in the paper we consider one of the simplest classes of extensions, namely, the class of so-called trivial extensions. It is shown that if a semigroup $L$ is a trivial extension of the semigroup $S$ by means of a group $G$, then there exists the embedding of the reduced semigroup $C^*$-algebra $C^*_r(S)$ into the $C^*$-algebra $C^*_r(L)$ which is induced by an embedding of the semigroup $S$ into the semigroup $L$.
In the work we also introduce and study the structure of a Banach $C^*_r(S)$-module on the underlying space of the reduced semigroup $C^*$-algebra $C^*_r(L)$. To do this, we use a topological grading for the $C^*$-algebra $C^*_r(L)$ over the group $G$. In the case when a semigroup $L$ is a trivial extension of a semigroup $S$ by means of a finite group, we prove the existence of the structure of a free Banach module over the reduced semigroup $C^*$-algebra $C^*_r(S)$ on the underlying Banach space of the semigroup $C^*$-algebra $C^*_r(L)$.
We give examples of extensions of semigroups and reduced semigroup $C^*$-algebras for a more complete characterization of the issues under consideration and for revealing connections with previous results.