Consider a Banach space $(\mathcal{A},\|\cdot\|_{\mathcal{A}}$) and assume that $\mathcal{A}$ is also an associative algebra with a unit element $I\in \mathcal{A}$ such that $\|I\|_{\mathcal{A}}=1$. Furthermore, suppose that there exists a constant $C>0$ such that for all $A,B\in \mathcal{A}$ $$\|AB\|_{\mathcal{A}}\leq C\|A\|_{\mathcal{A}}\|B\|_{\mathcal{A}}. $$
It's obvious (see e.g. this post) that one can rescale the norm to obtain an equivalent norm that is submultiplicative. However, in this case the identity element will no longer have norm one.
Question: How can we define an equivalent norm $\|\cdot\|_{\mathcal{A}}'$ on $\mathcal{A}$ such that this new norm is submultiplicative and $\|I\|_{\mathcal{A}}'=1$?
Define $$\|A\|'=\sup\{\|AB\| : \|B\|\leq 1\}.$$