# Maximal abelian subalgebras of $B\left(\mathcal{X}\right)$

Commentationes Mathematicae (2008)

- Volume: 48, Issue: 1
- ISSN: 2080-1211

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topJanko Bračič, and Bojan Kuzma. "Maximal abelian subalgebras of $B(\mathcal {X})$." Commentationes Mathematicae 48.1 (2008): null. <http://eudml.org/doc/291417>.

@article{JankoBračič2008,

abstract = {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 $.},

author = {Janko Bračič, Bojan Kuzma},

journal = {Commentationes Mathematicae},

keywords = {Abelian algebra; Bounded operators; Complex Banach space},

language = {eng},

number = {1},

pages = {null},

title = {Maximal abelian subalgebras of $B(\mathcal \{X\})$},

url = {http://eudml.org/doc/291417},

volume = {48},

year = {2008},

}

TY - JOUR

AU - Janko Bračič

AU - Bojan Kuzma

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

JO - Commentationes Mathematicae

PY - 2008

VL - 48

IS - 1

SP - null

AB - 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 $.

LA - eng

KW - Abelian algebra; Bounded operators; Complex Banach space

UR - http://eudml.org/doc/291417

ER -

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