What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$?

Question

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?

Answer 1

The only example is $\Bbb C$. Say $A$ is a Banach algebra with the given property and say the identity is $e$.

First suppose that $a$ is invertible and $\lambda\in\sigma(a)$, the spectrum of $a$. Then $|\lambda|\le||a||$. And $\lambda^{-1}\in\sigma(a^{-1})$, so $|\lambda^{-1}|\le||a^{-1}||=||a||^{-1}$. So $|\lambda|=||a||$.

Now if $a$ is any element of $A$, say $\lambda$ is a boundary point of $\sigma(a)$, and choose $z_n\in\Bbb C\setminus\sigma(a)$ with $z_n\to\lambda$. Then $a-z_ne$ is invertible and $\lambda-z_n\in\sigma(a-z_ne)$, so $$||a-\lambda e||=\lim||a-z_ne||=\lim|\lambda-z_n|=0.$$Hance $A=\{\lambda e:\lambda\in\Bbb C\}$.