## Functiones et Approximatio Commentarii Mathematici

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

Wiesław Żelazko

#### Abstract

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

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

Dates
First available in Project Euclid: 17 December 2015

https://projecteuclid.org/euclid.facm/1450389051

Digital Object Identifier
doi:10.7169/facm/2015.53.2.2

Mathematical Reviews number (MathSciNet)
MR3435796

Zentralblatt MATH identifier
1093.53082

#### Citation

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. https://projecteuclid.org/euclid.facm/1450389051

#### References

• I.M. Gelfand, Normierte Ringe, Mat. Sb. 9 (1941), 3–24.
• N.J. Kalton, J.W. Roberts, A rigid subspace of $L_0$, Trans. Amer. Math. Soc. 266 (1981), 645–654.
• B. Mityagin, S. Rolewicz, W. Żelazko, Entire functions in $B_0$-algebras, Studia Math. 21 (1962), 291–306.
• S. Rolewicz, Metric Linear Spaces, PWN-Reidel, Warszawa, Dordrecht 1984.
• W. Żelazko, An infinite dimensional Banach algebra with all but one maximal abelian subalgebras of dimension two, Comm. Math. 48 (2008), 99–101.