All Schatten spaces endowed with the Schur product are \(Q\)-algebras
type: Item
instance of: scholarly article
title: All Schatten spaces endowed with the Schur product are \(Q\)-algebras
zbMATH Open document ID: 1237.46036
author: Jop Briët
author: Harry Buhrman
author: Troy Lee
author: Thomas Vidick
published in: Journal of Functional Analysis
publication date: 2011-12-14T00:00:00Z
review text: 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.
reviewed by: Mati Abel
Mathematics Subject Classification ID: 46H35
zbMATH DE Number: 5988389
zbMATH Keywords: Scatten space
zbMATH Keywords: Schur multiplication
zbMATH Keywords: commutative Banach algebra
zbMATH Keywords: \(Q\)-algebra
MaRDI profile type: Publication
Wikidata QID: Q59792760
full work available at URL: j.jfa.2011.09.001
OpenAlex ID: W2120442492
cites work: Q4112938
cites work: Multidimensional extensions of the Grothendieck inequality and applications
cites work: Banach Lattices and Extensions of Grothendieck's Inequality
cites work: Quotient Algebras of Uniform Algebras
cites work: Q4850055
cites work: Q3239535
cites work: A new upper bound for the complex Grothendieck constant
cites work: Absolutely summing operators in $ℒ_{p}$-spaces and their applications
cites work: The Schatten space 𝑆₄ is a 𝑄-algebra
cites work: Q5483187
cites work: Unbounded violation of tripartite Bell inequalities
cites work: Interpolationstheorie für Banachideale von beschränkten linearen Operatoren
cites work: The Von Neumann Inequality for Polynomials in Several Hilbert-Schmidt Operators
cites work: Some remarks on Q-algebras
cites work: On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. (Appendix by S. Kaijser and N. Th. Varopoulos.)
cites work: A theorem on operator algebras.
cites work: Q5566034
DOI: 10.1016/J.JFA.2011.09.001
Recommended article: The Schatten space 𝑆₄ is a 𝑄-algebra
Recommended article: Absolute Schur Algebras and Unbounded Matrices
Recommended article: Q3700298
Recommended article: Q5483187
Recommended article: Q3988933
Recommended article: Bilinear mappings and trace class ideals
Recommended article: Wiener's lemma for infinite matrices of Gohberg-Baskakov-Sjöstrand class
Recommended article: Compact Endomorphisms and Closed Ideals in Banach Algebras
Recommended article: Uniqueness of the Uniform Norm with an Application to Topological Algebras
Recommended article: A System of Operator Equations
description: scientific article; zbMATH DE number 5988389