Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T08:28:38.605Z Has data issue: false hasContentIssue false
  • English
  • Français

REDUCED PRODUCTS OF METRIC STRUCTURES: A METRIC FEFERMAN–VAUGHT THEOREM

Published online by Cambridge University Press:  14 September 2016

SAEED GHASEMI
SAEED GHASEMI*
Affiliation:
INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES UL. ŚNIADECKICH 8, 00-656 WARSZAWA, POLANDE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the classical Feferman–Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form ⊕iMk(i) (ℂ) such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely noncommutative coronas of separable C*-algebras with this property.

Keywords

reduced productsmetric model theorymetric Feferman-Vaught theoremcorona algebrastrivial isomorphisms
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 856 - 875
Copyright
Copyright © The Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Akemann, C. A., Pedersen, G. K., and Tomiyama, J., Multipliers of C*-algebras . Journal of Functional Analysis, vol. 13 (1973), no. 3, pp. 277–301.CrossRefGoogle Scholar
Ben Yaacov, I., Berenstein, A., Henson, C.W., and Usvyatsov, A., Model theory for metric structures , Model Theory with Applications to Algebra and Analysis, Vol. II (Chatzidakis, Z. et al. ., editors), London Mathematical Society Lecture Note Series, no. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427.Google Scholar
Blackadar, B., Operator algebras , Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006.Google Scholar
Brown, N. P., On quasidiagonal C*-algebras , Operator Algebras and Applications, Advanced Studies in Pure Mathematics, vol. 38, Mathematical Society of Japan, Tokyo, 2004, pp. 19–64.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory, third ed., Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, vol. 73, 1990.Google Scholar
Dow, A. and Hart, K. P., ω * has (almost) no continuous images . Israel Journal of Mathematics, vol. 109 (1999), pp. 29–39.CrossRefGoogle Scholar
Farah, I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers . Memoirs of American Mathematical Society, vol. 147 (2000), no. 702.Google Scholar
Farah, I., All automorphisms of the Calkin algebra are inner . Annals of Mathematics, vol. 173 (2011), pp. 619–661.CrossRefGoogle Scholar
Farah, I., How many Boolean algebras ${\cal P}$ (ℕ) / ${\cal I}$ are there? Illinois Journal of Mathematics, vol. 46 (2003), pp. 999–1033.Google Scholar
Farah, I., Logic and operator algebras , Proceedings of the Seoul ICM, vol. II (2014), pp. 15–39.Google Scholar
Farah, I. and Hart, B., Countable saturation of corona algebras . Comptes Rendus Mathématiques, vol. 35 (2013), no. 2, pp. 35–56.Google Scholar
Farah, I., Hart, B., Lupini, M., Robert, L., Tikuisis, A., Vignati, A. and Winter, W., Model theory of nuclear C*-algebras, preprint, 2016, arXiv:1602.08072v1.Google Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras II: Model theory . Israel Journal of Mathematics, vol. 201 (2014), pp. 477–505.CrossRefGoogle Scholar
Farah, I., and Shelah, S., Trivial automorphisms . Israel Journal of Mathematics, vol. 201 (2014), no. 2, pp. 701–728.CrossRefGoogle Scholar
Farah, I., and Shelah, S., Rigidity of continuous quotients . Journal of the Institute of Mathematics of Jussieu, vol. 15 (2016), pp. 1–28.CrossRefGoogle Scholar
Feferman, S., Vaught, R. L., The first order properties of products of algebraic systems . Fundamenta Mathematicae, T. XLVII, pp. 57–103, 1959.CrossRefGoogle Scholar
Frayne, T., Morel, A. C. and Scott, D. S., Reduced direct products . Fundamenta Mathematicae, vol. 51 (1962), pp. 195–228.CrossRefGoogle Scholar
Ghasemi, S., Isomorphisms of quotients of FDD-algebras . Israel Journal of Mathematics, vol. 209 (2015), no. 2, pp. 825–854.CrossRefGoogle Scholar
Hart, B., Continuous model theory and its applications, 2012, Course notes, available at http://www.math.mcmaster.ca/∼bradd/courses/math712/index.html.Google Scholar
Lopes, V. C., Reduced products and sheaves of metric structures . Mathematical Logic Quarterly, vol. 59 (2013), no. 3, pp. 219–229.CrossRefGoogle Scholar
Mazur, K., F σ -ideals and $\omega _1 \omega _1^{\rm{*}} $ -gaps in the Boolean algebra P (ω)/ℐ. Fundamenta Mathematicae, vol. 138 (1991), pp. 103–111.CrossRefGoogle Scholar
Parovičenko, I. I., A universal bicompact of weight ℵ. Soviet Mathematics Doklady, vol. 4 (1963), pp. 592–592. Ob odnom universal’nom bikompakte vesa ℵ. Doklady Akademii Nauk SSSR , vol. 150(1963), pp. 36–39 (Russian).Google Scholar
Phillips, N. C. and Weaver, N., The Calkin algebra has outer automorphisms . Duke Mathematical Journal, vol. 139 (2007), pp. 185–202.CrossRefGoogle Scholar
Solecki, S., Analytic ideals and their applications . Annals of Pure and Applied Logic, vol. 99 (1999), pp. 51–72.CrossRefGoogle Scholar
Voiculescu, D., A note on quasi-diagonal C*-algebras and homotopy. Duke Mathematical Journal, vol. 62 (1991), no. 2, pp. 267–271.CrossRefGoogle Scholar
Vourtsanis, Y., A direct droof of the Feferman-Vaught theorem and other preservation theorems in products, this Journal, vol. 56 (1991), no. 2, pp. 632–636.Google Scholar