On the complete boundedness of the Schur block product

2017/12/14 by Christensen, Erik
#15A69 #46L07 #81P68 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1712.05285

Abstract

We give a Stinespring representation of the Schur block product, say (*), on pairs of square matrices with entries in a C*-algebra as a completely bounded bilinear operator of the form: A:=(a_ij), B:= (b_ij): A (*) B := (a_ijb_ij) = V* pi(A) F pi(B) V, such that V is an isometry, pi is a *-representation and F is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new ones on diagonals of matrices. ||A (*) B|| ≤ ||A||_r ||B||_c operator, row and column norm; - diag(A*A) ≤ A* (*) A ≤ diag(A*A), and for all vectors f, g: | |² ≤ < diag(AA*) g, g> .

Related

• Bilinear forms, Schur multipliers, complete boundedness and duality
• Decompositions of Schur block products
• Schur product with operator-valued entries
• Commutator inequalities via Schur products
• Recursive projections of symmetric tensors and Marcus's proof of the Schur inequality