On a condition similar to the star algebra

Question

Recently I have been reading about star algebras. In particular, $C^\ast$-algebras. It seems that the condition $\|a^\ast a\| = \|a\|^2$ is quite strong and much is known about $C^\ast$-algebras.

I was wondering if it makes sense to consider the class of algebras satisfying the condition

$$ \|a^{-1}\| = \|a\|^{-1}$$

or, similarly/alternatively the condition

$$\|a^{-1}\|=\|a\|$$

DOes either one of these two make for an interesting structure to be studied?

Answer 1

Both conditions are way too strong. In the case of continuous functions $C(X)$ on a compact Hausdorff space, the first condition implies that $X$ is a point. The second condition is just never satisfied (take $a=2$). – Qiaochu Yuan yesterday

If the first property holds for all invertible operators in the algebra of bounded operators on a Banach space, then the space is one dimensional. If it holds for a single element, that element is a scalar multiple of an isometry.