Geodesic
Maximal abelian subalgebras of \(B(\mathcal{X})\)
Commentationes Mathematicae, Tome 48 (2008) no. 1, pp. 95-98.

Voir la notice de l'article provenant de la source Annales Societatis Mathematicae Polonae Series

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 of \(B(\mathcal{X})\) 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 \(B(\mathcal{X})\) if \(\text{dim} X = \infty\).
DOI : 10.14708/cm.v48i1.5262
Mots-clés : Abelian algebra, Bounded operators, Complex Banach space 
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Janko Bračič; Bojan Kuzma. Maximal abelian subalgebras of \(B(\mathcal{X})\). Commentationes Mathematicae, Tome 48 (2008) no. 1, pp.  95-98. doi : 10.14708/cm.v48i1.5262. http://geodesic.mathdoc.fr/articles/10.14708/cm.v48i1.5262/

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