All Schatten spaces endowed with the Schur product are \(Q\)-algebras
Let \(S_\infty\) denote the space of compact operators \(A\) on \(l_2\) such that \[ A=\sum _{n=1}^\infty\lambda _n\langle \cdot, e_n\rangle f_n, \] where \((e_n)\) and \( (f_n)\) are orthonormal bases for \(l_2\) and \((\lambda _n)\subset \mathbb{R}\) satisfies the conditions \(\lambda _1\geqslant \lambda _2 \geqslant \cdots \geqslant 0\) and \(\lim _{n\to \infty}\lambda _n =0\). The space \(S_\infty\) is endowed with the usual norm \(\|A\|= \sup \{|\langle x, Ay\rangle |: \|x\|, \|y\|\leqslant 1\}.\) Moreover, for each \(1\leqslant p <\infty \) let \(\|A\|_p=(\mathrm{Tr}|A|^p)^\frac{1}{p}\), where \(|A|= (A^*A)^\frac{1}{2}\), and \(S_p\) be the subspace of elements \(A\in S_\infty\) such that \(\|A\|_p\) is finite. With respect to the Schur multiplication, \(S_p\) with \(1\leqslant p\leqslant \infty\) is a commutative Banach algebra. In [Math. Scand. 37, 173--182 (1975; Zbl 0337.46041)] \textit{N. Th. Varopoulos} asked: Is it true that \(S_p\) is a \(Q\)-algebra for all \(1\leqslant p \leqslant\infty\)? (A commutative Banach algebra \(A\) is a \(Q\)-algebra if there exists a uniform algebra \(B\) and a closed ideal \(I\) of \(B\) such that \(A\) is isomorphic, as a Banach algebra, to \(B/I\).) A positive answer to this problem is given in the present paper.
scientific article; zbMATH DE number 5988389
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