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arXiv · 1712.05285

On the complete boundedness of the Schur block product

Erik Christensen
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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_{ij}b_{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|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(A*A) \leq A* (*) A \leq diag(A*A), and for all vectors f, g: | |^2 \leq < diag(AA*) g, g> .

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Erik Christensen. 2017-12-14. On the complete boundedness of the Schur block product. https://arxiv.org/abs/1712.05285

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