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

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.