Maximal abelian subalgebras of \(B(\mathcal{X})\)

• Autorzy: Janko Bračič , Bojan Kuzma
• Języki publikacji: EN

Abstrakty

EN

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 \(\text{dim} X = \infty\).

Słowa kluczowe

EN

Abelian algebra; Bounded operators; Complex Banach space

• Wydawca: Polish Mathematical Society
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2008
• Tom: 48
• Numer: 1

Daty

wydano

2008

online

2017-12-19

Twórcy

autor

Janko Bračič

autor

Bojan Kuzma

Identyfikatory

DOI

10.14708/cm.v48i1.5262