semigroups defined on banach algebra

Question

Let $(X,\|.\|)$ be a Banach space and $Z:=\{Z(t)\}_{t\geq 0}$ is strongly continuous semigroup defined on it. If $X$ turns out to be a Banach Algebra, i.e. for $x,y\in X$, $xy\in X$. Is $Z$ still defined on $X$? If yes, how $Z(t)$ will operate on the product $xy$? As I expect it to be like $$Z(t)[xy]= [Z(t)x][Z(t)y]$$ It would be great if someone can provide me with proper reference.