In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II₁ factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ₀-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 II₁ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-3-1,
author = {David Sherman},
title = {Notes on automorphisms of ultrapowers of II1 factors},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {201-217},
zbl = {1183.46067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-3-1}
}
David Sherman. Notes on automorphisms of ultrapowers of II₁ factors. Studia Mathematica, Tome 192 (2009) pp. 201-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-3-1/