Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T10:11:20.693Z Has data issue: false hasContentIssue false
  • English
  • Français

FRAÏSSÉ LIMITS OF C*-ALGEBRAS

Published online by Cambridge University Press:  29 June 2016

CHRISTOPHER J. EAGLE ,
ILIJAS FARAH ,
BRADD HART ,
BORIS KADETS ,
VLADYSLAV KALASHNYK  and
MARTINO LUPINI
CHRISTOPHER J. EAGLE
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO 40 ST. GEORGE STREET TORONTO, ONTARIO, M5S 2E4CANADAE-mail:[email protected]
ILIJAS FARAH
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS YORK UNIVERSITY4700KEELE STREET NORTH YORK, ONTARIO, M3J 1P3CANADA MATEMATICKI INSTITUT KNEZA MIHAILA 34 BELGRADE, SERBIAE-mail:[email protected]: http://www.math.yorku.ca/∼ifarah
BRADD HART
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS MCMASTER UNIVERSITY1280MAIN STREET WEST HAMILTON, ONTARIO, L8S 4K1CANADAE-mail:[email protected]: http://www.math.mcmaster.ca/∼bradd/
BORIS KADETS
Affiliation:
SCHOOL OF MATHEMATICS AND MECHANICAL ENGINEERING KHARKIV V.N. KARAZIN NATIONAL UNIVERSITY KHARKIV, UKRAINEE-mail:[email protected]
VLADYSLAV KALASHNYK
Affiliation:
DEPARTMENT OF MATHEMATICS 340 ROWLAND HALL UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA92697-3875, USAE-mail:[email protected]
MARTINO LUPINI
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS YORK UNIVERSITY4700KEELE STREET NORTH YORK, ONTARIO, M3J 1P3CANADAE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II1 factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.

Keywords

Fraïssé limitsAF algebrasJiang-Su algebralogic of metric structures
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 755 - 773
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

Ben Yaacov, I., Fraïssé limits of metric structures, this Journal, vol. 80 (2015), no. 1, pp. 100–115.Google 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, vol. 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
Davidson, K., C*-Algebras by Example, American Mathematical Society, Providence, RI, 1996.CrossRefGoogle Scholar
Eagle, C. J., Farah, I., Kirchberg, E., and Vignati, A., Quantifier elimination in C*-algebras, (2015). arXiv:1502.00573.Google Scholar
Eagle, C. J., Goldbring, I., and Vignati, A., The pseudoarc is a co-existentially closed continuum, (2015). arXiv:1503.03443.Google Scholar
Eagle, C. J. and Vignati, A., Saturation and elementary equivalence of C*-algebras . Journal of Functional Analysis, vol. 269 (2015), pp. 2631–2664.Google Scholar
Elliott, G. A. and Toms, A. S., Regularity properties in the classification program for separable amenable C*-algebras . Bulletin of the American Mathematical Society, vol. 45 (2008), no. 2, pp. 229–245.Google Scholar
Farah, I., Logic and operator algebras , Proceedings of the Seoul ICM (Jang, S. Y., Kim, Y. R., Lee, D.-W., and Yie, I., editors), vol. II, Kyung Moon SA, Seoul, 2014, pp. 15–40.Google Scholar
Farah, I., Selected applications of logic to classification problem of C*-algebras , E-recursion, Forcing and C*-Algebras (Chong, C. T. et al. , editors), Lecture Note Series, Institute for Mathematical Sciences, National University of Singapore, vol. 27, World Scientific, Singapore, 2014, pp. 1–82.Google Scholar
Farah, I., Hart, B., Rørdam, M., and Tikuisis, A., Relative commutants of strongly self-absorbing C*-algebras, Selecta Mathematica , to appear.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.Google Scholar
Farah, I., Toms, A. S., and Törnquist, A., Turbulence, orbit equivalence, and the classification of nuclear C*-algebras . Journal für die reine und angewandte Mathematik, vol. 688 (2014), pp. 101–146.CrossRefGoogle Scholar
Fraïssé, R., Sur l’extension aux relations de quelques propriétés des ordres . Annales Scientifiques de l’École Normale Supérieure. Troisième Série, vol. 71 (1954), pp. 363–388.Google Scholar
Giordano, T. and Pestov, V., Some extremely amenable groups . Comptes Rendus Mathematique, vol. 334 (2002), no. 4, pp. 273–278.CrossRefGoogle Scholar
Glimm, J. G., On a certain class of operator algebras . Transactions of the American Mathematical Society, vol. 95 (1960), no. 2, pp. 318–340.Google Scholar
Gromov, M., Filling Riemannian manifolds . Journal of Differential Geometry, vol. 18 (1983), no. 1, pp. 1–147.Google Scholar
Hart, B., Goldbring, I., and Sinclair, T., The theory of tracial von Neumann algebras does not have a model companion, this Journal, vol. 78 (2013), no. 3, pp. 1000–1004.Google Scholar
Jacelon, B., A simple, monotracial, stably projectionless C*-algebra . Journal of the London Mathematical Society, vol. 87 (2013), pp. 365–383.CrossRefGoogle Scholar
Jiang, X. and Su, H., On a simple unital projectionless C*-algebra . American Journal of Mathematics, vol. 121 (1999), pp. 359–413.Google Scholar
Kechris, A. S., Pestov, V., and Todorcevic, S., Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups . Geometric & Functional Analysis GAFA, vol. 15 (2005), no. 1, pp. 106–189.CrossRefGoogle Scholar
Kirchberg, E., Exact C*-algebras, tensor products, and the classification of purely infinite algebras , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, , 1994), Birkhäuser, Basel, 1995, pp. 943–954.Google Scholar
Kubiś, W. and Solecki, S., A proof of uniqueness of the Gurariĭ space . Israel Journal of Mathematics, vol. 195 (2013), no. 1, pp. 449–456.Google Scholar
Lupini, M., Uniqueness, universality, and homogeneity of the noncommutative Gurarij space, (2014). arXiv:1410.3345.Google Scholar
Masumoto, S., Jiang-Su algebra as a Fraïssé limit, (2016). arXiv:1602.00124.Google Scholar
Melleray, J. and Tsankov, T., Extremely amenable groups via continuous logic, (2014). arxiv:1404.4590.Google Scholar
Pestov, V., Dynamics of Infinite-Dimensional Groups, University Lecture Series, vol. 40, American Mathematical Society, Providence, RI, 2006.Google Scholar
Razak, S., On the classification of simple stably projectionless C*-algebras . Canadian Journal of Mathematics, vol. 54 (2002), no. 1, pp. 138–224.Google Scholar
Robert, L., Classification of inductive limits of 1-dimensional NCCW complexes . Advances in Mathematics, vol. 231 (2012), no. 5, pp. 2802–2836.Google Scholar
Rørdam, M., Larsen, F., and Laustsen, N. J., An Introduction to K-Theory for C* Algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000.Google Scholar
Rørdam, M. and Størmer, E., Classification of nuclear C*-algebras. Entropy in operator algebras , Encyclopaedia of Mathematical Sciences, vol. 126, Operator Algebras and Non-commutative Geometry, vol. 7, Springer-Verlag, Berlin, 2002.Google Scholar
Sabok, M., Completeness of the isomorphism problem for separable C*-algebras . Inventiones Mathematicae, (2015), pp.1–36.Google Scholar
Sato, Y., White, S., and Winter, W., Nuclear dimension and Ƶ-stability . Inventiones Mathematicae, vol. 202 (2015), pp. 893–921.CrossRefGoogle Scholar
Schochet, C., Topological methods for C*-algebras II: Geometric resolutions and the Künneth formula . Pacific Journal of Mathematics, vol. 98 (1982), pp. 399–445.Google Scholar
Schoretsanitis, K., Fraïssé Theory for Metric Structures , Ph.D. thesis, University of Illinois at Urbana-Champaign, 2007.Google Scholar
Thiel, H. and Winter, W., The generator problem for Ƶ-stable C*-algebras . Transactions of the American Mathematical Society, vol. 366 (2014), pp. 2327–2343.CrossRefGoogle Scholar
Toms, A. S. and Winter, W., Strongly self-absorbing C*-algebras . Transactions of the American Mathematical Society, vol. 359 (2007), no. 8, pp. 3999–4029.CrossRefGoogle Scholar