Checking whether $\mathbb C^n$ is semisimple or not

Question

Consider $\mathbb C^n$ as a Banach algebra over $\mathbb C$. Is it semisimple?

A Banach algebra is semisimple if the intersection of all maximal ideals of it is ${0}$.

I can't figure out what are the maximal ideals of $\mathbb C^n$. How do I solve it?

score 1 · Accepted Answer · answered Jul 31 '20 at 15:22

1

What about putting $I_k:=\{x\in\mathbb{C}^n\mid x_k=0\}$?

They are surely ideals, they are maximal since their dimension is $n-1$, and their intersection is zero.

answered Jul 31 '20 at 15:22

Daniel

One more way to answer this : $\mathbb C ^n$ is a $C^$ algebra and every $C^$ algebra is semisimple. – Shreya Chauhan Oct 30 '20 at 02:09

1 Answers1