C*-algebraic Schur product theorem, P\'{o}lya-Szeg\H{o}-Rudin question and Novak's conjecture

TL;DR

This paper extends the Schur product theorem to matrices over C*-algebras, proves a C*-algebraic version of Novak's conjecture for commutative cases, and explores related inequalities and questions.

Contribution

It introduces a C*-algebraic framework for the Schur product, proves a version of Novak's conjecture in this setting, and formulates the Pólya-Szegő-Rudin question for C*-algebraic Schur products.

Findings

C*-algebraic Schur product bounds established

Novak's conjecture proved for commutative unital C*-algebras

Formulation of Pólya-Szegő-Rudin question in C*-algebra context

Abstract

Striking result of Vyb\'{\i}ral [\textit{Adv. Math.} 2020] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vyb\'{\i}ral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vyb\'{\i}ral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate P\'{o}lya-Szeg\H{o}-Rudin question for the C*-algebraic Schur product of positive matrices.