In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra version of the aforementioned result exist?
That is, if $M$ is a closed left ideal of a Banach algebra $\mathcal{A}$, $F$ is a minimal left ideal of $\mathcal{A}$, whether or not $M+F$ is closed too?
In particular, let $a \in \mathcal{A}$, $R(a):=\{x\in \mathcal{A}: ax=0\}$. If $a\mathcal{A}$ is closed and $R(a)$ is minimal, whether or not $a\mathcal{A}+R(a)$ is closed?
We know that every closed Banach subspace is a Banach space for the induced norm and every finite-dimensional Banach subspace is also a Banach space. as the sum is stable so the sum is a banach space and therefore it is closed.
Mr. @Dauphin Mwami Kamwanga Wa Kamwanga:
Mr. @Qingping Zeng is talking about the closedness of the sum of a closed left ideal and a minimal left ideal of a Banach algebra.
Dear Professor Dauphin Mwami Kamwanga Wa Kamwanga，thank you for your interest in the question. The question is exactly:
if $M$ is a closed left ideal of a Banach algebra $\mathcal{A}$, $F$ is a minimal left ideal of $\mathcal{A}$, whether or not $M+F$ is closed too?
Any hints or references are welcome.
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