  We generalize Loewner'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 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.
