Equality of norm $||L_{xy}|| = ||L_x \cdot L_y||$?

Question

Let $\mathcal{A}$ be a Banach algebra. We consider, for every $x \in \mathcal{A}$, the operator $L_x: \mathcal{A} \rightarrow \mathcal{A}$, with $L_x(y)=xy$.

Is it true that $||L_{xy}|| = ||L_x \cdot L_y || \leq ||L_x|| \cdot ||L_y||$ ?

I try to show it through the definition of the operator norm but I can't find a way to do it.

Any help / advice would be grateful.

Thank you very much.