Abstract view
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From Matrix to Operator Inequalities
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Published:2011-04-06
Printed: Jun 2012
Terry A. Loring,
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A.
Abstract
We generalize Löwner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ 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: |
$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional
$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional
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