Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional.
Question. Does it follow that $A$ is finite dimensional?
Remark. Clearly, every element of $A$ is algebraic (i.e. annihilated by a polynomial) and thus has finite spectrum. If $A$ is semisimple, it therefore follows from a result of Kaplansky that $A$ is finite dimensional (Lemma 7 in "I. Kaplansky: Ring Isomorphisms of Banach Algebras (1954)". So the question is concerned with the non-semi-simple case.