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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.

The Schatten space 𝑆₄ is a 𝑄-algebra