The Schatten space 𝑆₄ is a 𝑄-algebra

Abstract

For any 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , let S p S_{p} denote the classical p p -Schatten space of operators on the Hilbert space ℓ 2 \ell _{2} . It was shown by Varopoulos (for p ≥ 2 p \geq 2 ) and by Blecher and the author (full result) that for any 1 ≤ p ≤ ∞ , S p 1 \leq p \leq \infty , S_{p} equipped with the Schur product is an operator algebra. Here we prove that S 4 S_{4} (and thus S p S_{p} for any 2 ≤ p ≤ 4 2 \leq p \leq 4 ) is actually a Q Q -algebra, which means that it is isomorphic to some quotient of a uniform algebra in the Banach algebra sense.