i ask for an example of non-simple Banach algebra. non-simple means that the intersection of all maximal ideals is not zero.
2026-04-27 15:44:31.1777304671
Example of non-simple Banach algebra
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Let $B$ be the space of upper triangular $2\times 2$ matrices (with the operator norm). Consider $x=\begin{bmatrix}0&1\\0&0\end{bmatrix}$. If $\lambda\ne0$ then $x-\lambda I$ is invertible; hence $x$ lies in every maximal ideal.