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

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.