The Schatten space $S_{4}$ is a $Q$-algebra

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