Can we define unitary representations on semigroups

Question

A representation on a semigroup $S$ is a pair $(\pi,H_\pi)$ where $\pi$ is a homomorphism from $S$ into $B(H_\pi)$ and $H_\pi$ is a Hilbert space.

In the group case, a representation $\pi$ of a group $G$ is said to be unitary if $\pi(g)$ is a unitary operator for all $g\in G$.

Can I use the same definition for a semigroup?

Answer 1

You are welcome to make the definition as you see fit. A unitary representation of a semigroup would have the same meaning as a unitary representation of a group.