The Schatten space -algebra

For any , let denote the classical -Schatten space of operators on the Hilbert space . It was shown by Varopoulos (for ) and by Blecher and the author (full result) that for any equipped with the Schur product is an operator algebra. Here we prove that (and thus for any ) is actually a -algebra, which means that it is isomorphic to some quotient of a uniform algebra in the Banach algebra sense.