From Matrix to Operator Inequalities
Abstract
We generalize Loewner's method for proving that matrix monotone functions are operator monotone. The relation x ≤ y on bounded operators is our model for a definition for C*-relations of 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.
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