On the complete boundedness of the Schur block product
Operator Algebras
2018-11-12 v3 Mathematical Physics
math.MP
Quantum Algebra
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: |<A(*)B f,g> |^2 \leq < diag(AA*) g, g> <diag(B*B) f,f> .
Keywords
Cite
@article{arxiv.1712.05285,
title = {On the complete boundedness of the Schur block product},
author = {Erik Christensen},
journal= {arXiv preprint arXiv:1712.05285},
year = {2018}
}
Comments
10 p, revised, expanded and to appear in Proc. AMS