This is an announcement for the paper "A dichotomy for the number of ultrapowers" by Ilijas Farah and Saharon Shelah.

Abstract: We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are $2^{2^{\aleph_0}}$ many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II$_1$ factors, as well as their relative commutants and include several applications. We also show that the C*-algebra B(H) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on N.

Archive classification: math.LO math.OA

Mathematics Subject Classification: 03C20, 46M07

Report Number: Shelah [FaSh:954]

The source file(s), 2009i19-ultrapowers.tex: 122804 bytes, is(are) stored in gzipped form as 0912.0406.gz with size 33kb. The corresponding postcript file has gzipped size 176kb.

Submitted from: ifarah@yorku.ca

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0912.0406

or

http://arXiv.org/abs/0912.0406

or by email in unzipped form by transmitting an empty message with subject line

uget 0912.0406

or in gzipped form by using subject line

get 0912.0406

to: math@arXiv.org.