Journal of the
Korean Mathematical Society JKMS

Striking result of Vyb\'{\i}ral \cite{VYBIRAL} 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.

Keywords: Schur/Hadamard product, positive matrix, Hilbert C*-module, C*-algebra, Schur product theorem, P\'{o}lya-Szeg\H{o}-Rudin question, Novak's conjecture

MSC numbers: Primary 15B48, 46L05, 46L08

Supported by: The author is supported by Indian Statistical Institute, Bangalore, through the J.~C.~Bose Fellowship of Prof.~B.~V.~Rajarama Bhat.