Skip to main content



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 ⊕ i M k(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.

Hide All
[1] Akemann, C. A., Pedersen, G. K., and Tomiyama, J., Multipliers of C*-algebras . Journal of Functional Analysis, vol. 13 (1973), no. 3, pp. 277301.
[2] 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. 315427.
[3] Blackadar, B., Operator algebras , Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006.
[4] 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. 1964.
[5] 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.
[6] Dow, A. and Hart, K. P., ω * has (almost) no continuous images . Israel Journal of Mathematics, vol. 109 (1999), pp. 2939.
[7] 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.
[8] Farah, I., All automorphisms of the Calkin algebra are inner . Annals of Mathematics, vol. 173 (2011), pp. 619661.
[9] Farah, I., How many Boolean algebras ${\cal P}$ (ℕ) / ${\cal I}$ are there? Illinois Journal of Mathematics, vol. 46 (2003), pp. 9991033.
[10] Farah, I., Logic and operator algebras , Proceedings of the Seoul ICM, vol. II (2014), pp. 1539.
[11] Farah, I. and Hart, B., Countable saturation of corona algebras . Comptes Rendus Mathématiques, vol. 35 (2013), no. 2, pp. 3556.
[12] 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.
[13] Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras II: Model theory . Israel Journal of Mathematics, vol. 201 (2014), pp. 477505.
[14] Farah, I., and Shelah, S., Trivial automorphisms . Israel Journal of Mathematics, vol. 201 (2014), no. 2, pp. 701728.
[15] Farah, I., and Shelah, S., Rigidity of continuous quotients . Journal of the Institute of Mathematics of Jussieu, vol. 15 (2016), pp. 128.
[16] Feferman, S., Vaught, R. L., The first order properties of products of algebraic systems . Fundamenta Mathematicae, T. XLVII, pp. 57103, 1959.
[17] Frayne, T., Morel, A. C. and Scott, D. S., Reduced direct products . Fundamenta Mathematicae, vol. 51 (1962), pp. 195228.
[18] Ghasemi, S., Isomorphisms of quotients of FDD-algebras . Israel Journal of Mathematics, vol. 209 (2015), no. 2, pp. 825854.
[19] Hart, B., Continuous model theory and its applications, 2012, Course notes, available at∼bradd/courses/math712/index.html.
[20] Lopes, V. C., Reduced products and sheaves of metric structures . Mathematical Logic Quarterly, vol. 59 (2013), no. 3, pp. 219229.
[21] 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. 103111.
[22] Parovičenko, I. I., A universal bicompact of weight. 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).
[23] Phillips, N. C. and Weaver, N., The Calkin algebra has outer automorphisms . Duke Mathematical Journal, vol. 139 (2007), pp. 185202.
[24] Solecki, S., Analytic ideals and their applications . Annals of Pure and Applied Logic, vol. 99 (1999), pp. 5172.
[25] Voiculescu, D., A note on quasi-diagonal C*-algebras and homotopy. Duke Mathematical Journal, vol. 62 (1991), no. 2, pp. 267271.
[26] Vourtsanis, Y., A direct droof of the Feferman-Vaught theorem and other preservation theorems in products, this Journal, vol. 56 (1991), no. 2, pp. 632636.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 32 *
Loading metrics...

Abstract views

Total abstract views: 172 *
Loading metrics...

* Views captured on Cambridge Core between 14th September 2016 - 21st May 2018. This data will be updated every 24 hours.