Let $A$ be a Banach algebra, $I$ be a closed two-sided ideal in $A$, and $J$ be a closed two-sided ideal in $I$ such that there is no ideal between $I$ and $J$. Can we see $dim(\frac{I}{J})<\infty$?


Y. DOMAR in "On the ideal structure of certain Banach algebras", proves a lemma like this question with the following difference:


$A$ is commutative, $J$ is a two-sided ideal in $A$ and he shows $dim(\frac{I}{J})=1$



Dimension of quotient of ideals
If we allow $A$ to be noncommutative then there are well-known natural counterexamples. For instance one can look at $A=B(E)$ for various Banach spaces $E$. I imagine that there should be counterexamples of the form $A=L^1(G)$ but I do not know this for sure.


There are "silly" commutative counterexamples obtained by defining $I$ to be an infinite-dimensional Banach space with zero product, $J=\{0\}$, and setting $A$ to be the unitization of $I$.


