Notes on automorphisms of ultrapowers of II$_1$ factors
Volume 195 / 2009
Abstract
In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of $ \hbox {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 $ \hbox {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.
