Functiones et Approximatio Commentarii Mathematici

All maximal commutative subalgebras occur in $L(X)$ uncountably many times

Wiesław Żelazko

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We show that for every Banach space $X, \dim X>1$, every maximal commutative subalgebra of $L(X)$ has uncountably many copies between maximal commutative subalgebras of $L(X)$. Answering to a question of Aleksander Pe{\l}czy\'nski, we show also that for an arbitrary infinite dimensional Banach space $X$ there are at least countably many multiplications making of $X$ a commutative unital Banach algebra.

Article information

Funct. Approx. Comment. Math., Volume 53, Number 2 (2015), 189-192.

First available in Project Euclid: 17 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Secondary: 46H10: Ideals and subalgebras 46J05: General theory of commutative topological algebras

algebra of Banach space operators maximal commutative subalgebra multiplications on Banach spaces


Żelazko, Wiesław. All maximal commutative subalgebras occur in $L(X)$ uncountably many times. Funct. Approx. Comment. Math. 53 (2015), no. 2, 189--192. doi:10.7169/facm/2015.53.2.2.

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