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2 Commentationes Mathematicae

2 2008

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Maximal abelian subalgebras of B(X)

Bračič J., Kuzma B.

Commentationes Mathematicae

2008

Vol. 48, [Z] 1

95-98

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 = ∞.

An infinite dimensional Banach algebra with all but one maximal abelian subalgebras of dimension two

Żelazko W.

Commentationes Mathematicae

2008

Vol. 48, [Z] 1

99-101

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.