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

MSC Classifications:

46L05, 47B99 show english descriptions General theory of $C^*$-algebras
None of the above, but in this section 46L05 - General theory of $C^*$-algebras
47B99 - None of the above, but in this section

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