  In functional analysis, approximative properties of an object become precise
in its ultrapower. We discuss this idea and its consequences for automorphisms
of II_1 factors. Here are some sample results: (1) an automorphism is
approximately inner if and only if its ultrapower is aleph_0-locally inner; (2)
the ultrapower of an outer automorphism is always outer; (3) for unital
*-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a
II_1 factor, equality of the induced traces implies unitary equivalence. All
statements are proved using operator algebraic techniques, but in the last
section of the paper we indicate how the underlying principle is related to
theorems of Henson's positive bounded logic.
