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From Matrix to Operator Inequalities

Published online by Cambridge University Press:  20 November 2018

Terry A. Loring
Terry A. Loring*
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A.e-mail: loring@math.unm.edu
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Abstract

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We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\,\le \,y$ on bounded operators is our model for a definition of ${{C}^{*}}$-relations being residually finite dimensional.

Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices.

Applications are shown regarding norms of exponentials, the norms of commutators, and “positive” noncommutative $*$-polynomials.

Keywords

46L0547B99C*-algebrasmatricesbounded operatorsrelationsoperator normordercommutatorexponentialresidually finite dimensional
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 339 - 350
Copyright
Copyright © Canadian Mathematical Society 2012

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