$||a||\leq \sup_{||b||\leq 1} ||ab||$ in a C*-algebra

Question

I would like to prove that, if $a$ is an element in a C*-algebra then $$\|a\|\leq \sup_{\|b\|\leq 1} \|ab\|$$

It is obvious if the algebra is unital. What if it is not?

Answer 1

In a $C^\ast$ algebra, this is easy. You have $\Vert a \Vert = \Vert a^\ast \Vert$ and (by definition of a $C^\ast$ algebra) you have

$$ \Vert a^\ast \Vert^2 = \Vert (a^\ast)^\ast a^\ast \Vert = \Vert a \cdot a^\ast\Vert, $$

so that taking $b = \frac{a^\ast}{\Vert a\Vert}$ yields your claim, since the estimate $\Vert ab \Vert \leq \Vert a \Vert \Vert b \Vert \leq \Vert a \Vert$ for all $\Vert b \Vert \leq 1$ is trivial.