C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture

Abstract

Striking result of Vyb\'ıral [\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\'ıral 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\'ıral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate Pólya-Szegő-Rudin question for the C*-algebraic Schur product of positive matrices.

arXiv comment13 Pages, 0 Figures, Pólya-Szegő-Rudin question and Novak's conjecture are formulated and partially solved