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Let \(\mathcal{X}\) be an infinite dimensional complex Banach space and \(B(\mathcal{X})\) be the Banach algebra of all bounded linear operators on \(\mathcal{X}\). Żelazko [1] posed the following question: Is it possible that some maximal abelian subalgebra of \(B(\mathcal{X})\) is finite dimensional? Interestingly, he was able to show that there does exist an infinite dimensional closed subalgebra...
I construct a unital closed subalgebra of \(L(H)\) with the property announced in the title. Moreover, for any two maxiamal abelian subalgebras of the algebra in question, their intersection consists only of scalar multiples of the unity.
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