Geodesic
From Matrix to Operator Inequalities
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 339-350

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DOI

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.
DOI : 10.4153/CMB-2011-063-8
Mots-clés : 46L05, 47B99, C*-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional 
Loring, Terry A. From Matrix to Operator Inequalities. Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 339-350. doi: 10.4153/CMB-2011-063-8
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     title = {From {Matrix} to {Operator} {Inequalities}},
     journal = {Canadian mathematical bulletin},
     pages = {339--350},
     year = {2012},
     volume = {55},
     number = {2},
     doi = {10.4153/CMB-2011-063-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-063-8/}
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