Let X be an infinite dimensional complex Banach space and B(X) be the Banach algebra of all bounded linear operators on X. Zelazko [1] posed the following question: Is it possible that some maximal abelian subalgebra of B(X) is finite dimensional? Interestingly, he was able to show that there does exist an infinite dimensional closed subalgebra of B(X) with all but one maximal abelian subalgebras of dimension two. The aim of this note is to give a negative answer to the original question and prove that there does not exist a finite dimensional maximal commutative subalgebra of B(X) if dimX = ∞.
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