No Access
A DICHOTOMY FOR THE NUMBER OF ULTRAPOWERS
https://doi.org/10.1142/S0219061310000936Cited by:16
Abstract
We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ≤ 2ℵ0 associated with nonprincipal ultrafilters on ℕ. They are either all isomorphic, or else there are 22ℵ0 many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II1 factors, as well as their relative commutants and include several applications. We also show that the CAF001-algebra 
always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on ℕ.
Keywords:
AMSC: Primary 03C20, Secondary 46M07
References
-
T. Bartoszynski and H. Judah , Set Theory: On the Structure of the Real Line ( A. K. Peters , 1995 ) . Crossref, Google Scholar - , Model Theory with Applications to Algebra and Analysis,
Lecture Notes Series of the London Mathematical Society II, eds.Z. Chatzidakis (Cambridge University Press, 2008) pp. 315–427. Crossref, Google Scholar - I. Ben Ya'acov, W. Henson, M. Junge and Y. Raynaud, Preliminary report — vNA and NCP, preprint (2008) . Google Scholar
-
B. Blackadar , Operator Algebras ,Encyclopaedia of Mathematical Sciences 122 ( Springer-Verlag , Berlin , 2006 ) . Crossref, Google Scholar -
C. C. Chang and H. J. Keisler , Model Theory , 3rd edn. ,Studies in Logic and the Foundations of Mathematics 73 ( North-Holland Publishing Co. , Amsterdam , 1990 ) . Google Scholar - Topol. Proc. 9(2), 269 (1984). Google Scholar
- RIMS Kokyuroku 1819, 1 (2008). Google Scholar
- J. Funct. Anal. 256, 3841 (2009). Crossref, ISI, Google Scholar
- I. Farah, B. Hart and D. Sherman, Model theory of operator algebras I: Stability, preprint (2009) , arXiv:0908.2790 . Google Scholar
- I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, preprint (2009) , arXiv:1004.0741 . Google Scholar
- Adv. Math. 225, 1399 (2010). Crossref, ISI, Google Scholar
- I. Farah and T. Katsura, Nonseparable UHF algebras II: Classification, in preparation (2010) . Google Scholar
- Math. Ann. 347, 839 (2010). Crossref, ISI, Google Scholar
L. Ge and D. Hadwin , Ultraproducts of C*-algebras, Recent Advances in Operator Theory and Related Topics (1999)127,Operator Theory: Advances and Applications (Birkhäuser, Basel, 2001) pp. 305–326. Google Scholar- Adv. Math. 193, 142 (2005). Crossref, ISI, Google Scholar
-
K. Kunen , Set Theory: An Introduction to Independence Proofs ( North–Holland , 1980 ) . Google Scholar - P. Lücke and S. Thomas, Automorphism groups of ultraproducts of finite symmetric groups, preprint (2009) . Google Scholar
- Bull. Symb. Log. 14, 449 (2008). Crossref, ISI, Google Scholar
- S. Shelah, Non–Structure Theory, accepted (Oxford University Press) . Google Scholar
-
S. Shelah , Classification Theory and the Number of Nonisomorphic Models ,Studies in Logic and the Foundations of Mathematics 92 ( North-Holland Publishing Co. , Amsterdam , 1990 ) . Google Scholar - S. Shelah, General non-structure theory, preprint (2010) , arXiv: 1011.3576 . Google Scholar
- Israel J. Math. 166, 61 (2008). Crossref, ISI, Google Scholar
- S. Shelah and A. Usvyatsov, Unstable classes of metric structures, preprint . Google Scholar
- S. Thomas, On the number of universal sofic groups, preprint (2009) . Google Scholar
