Maximal abelian subalgebras of $B\left(𝒳\right)$

Janko 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 -