Notes on automorphisms of ultrapowers of II$_1$ factors

David Sherman¹

¹ Department of Mathematics University of Virginia P.O. Box 400137 Charlottesville, VA 22904, U.S.A.

Studia Mathematica, Tome 195 (2009) no. 3, pp. 201-217

Résumé

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.

Détail

DOI : 10.4064/sm195-3-1

Keywords: functional analysis approximative properties object become precise its ultrapower discuss idea its consequences automorphisms hbox factors here sample results automorphism approximately inner only its ultrapower aleph locally inner ultrapower outer automorphism always outer unital * homomorphisms separable nuclear * algebra ultrapower hbox factor equality induced traces implies unitary equivalence statements proved using operator algebraic techniques section paper indicate underlying principle related theorems hensons positive bounded logic