Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T22:59:18.979Z Has data issue: false hasContentIssue false
  • English
  • Français

THE CLASSIFICATION PROBLEM FOR AUTOMORPHISMS OF C*-ALGEBRAS

Published online by Cambridge University Press:  15 January 2016

MARTINO LUPINI
MARTINO LUPINI*
Affiliation:
MATHEMATICS DEPARTMENT CALIFORNIA INSTITUTE OF TECHNOLOGY 1200 E. CALIFORNIA BLVD MC 253-37 PASADENA, CA 91125, USAE-mail: [email protected]: http://www.lupini.org/
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.

Keywords

Primary 03E1546L40Secondary 46L3546L57C*-algebraautomorphismBorel complexityBorel reductionturbulenceCuntz algebra
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 21 , Issue 4 , December 2015 , pp. 402 - 424
Copyright
Copyright © The Association for Symbolic Logic 2015 

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, Charles A. and Pedersen, Gert K., Central sequences and inner derivations of separable C*-algebras. American Journal of Mathematics, vol. 101 (1979), no. 5, pp. 1047–1061.CrossRefGoogle Scholar
Argerami, Martín, Coskey, Samuel, Kalantar, Mehrdad, Kennedy, Matthew, Lupini, Martino, and Sabok, Marcin, The classification problem for finitely generated operator systems and spaces, arXiv:1411.0512, 2014.Google Scholar
Atiyah, Michael F. and Hirzebruch, Friedrich, Riemann-Roch theorems for differentiable manifolds. Bulletin of the American Mathematical Society, vol. 65 (1959), no. 4, pp. 276–281.CrossRefGoogle Scholar
Barlak, Selçuk and Szabó, Gabór, Rokhlin actions of finite groups on UHF-absorbing C*-algebras. Transactions of the American Mathematical Society, in press.Google Scholar
Yaacov, Itaï Ben, Berenstein, Alexander, Ward Henson, C., and Usvyatsov, Alexander, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, 2008, p. 315–427.Google Scholar
Yaacov, Itaï Ben, Nies, Andre, and Tsankov, Todor, A Lopez-Escobar theorem for continuous logic, arXiv:1407.7102, 2014.Google Scholar
Blackadar, Bruce, K-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986.Google Scholar
Blackadar, Bruce, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006.Google Scholar
Bosa, Joan, Brown, Nathanial P., Sato, Yasuhiko, Tikuisis, Aaron, White, Stuart, and Winter, Wilhelm, Covering dimension of C*-algebras and 2-coloured classification, arXiv:1506.03974, 2015.Google Scholar
Bratteli, Ola, Inductive limits of finite dimensional C*-algebras. Transactions of the American Mathematical Society, vol. 171 (1972), pp. 195–234.Google Scholar
Brown, Nathanial P. and Ozawa, Narutaka, C*-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, 2008.Google Scholar
Camerlo, Riccardo and Gao, Su, The completeness of the isomorphism relation for countable Boolean algebras. Transactions of the American Mathematical Society, vol. 353 (2001), no. 2, pp. 491–518.CrossRefGoogle Scholar
Coskey, Samuel and Lupini, Martino, A López-Escobar theorem for metric structures, and the topological Vaught conjecture, arXiv:1405.2859, 2014.Google Scholar
Cuntz, Joachim, Simple C*-algebras generated by isometries. Communications in Mathematical Physics, vol. 57 (1977), no. 2, pp. 173–185.CrossRefGoogle Scholar
Cuntz, Joachim, K-theory for certain C*-algebras. Annals of Mathematics, vol. 113 (1981), no. 1, pp. 181–197.CrossRefGoogle Scholar
Davidson, Kenneth R., C*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, 1996.Google Scholar
Downey, Rod and Montalbán, Antonio, The isomorphism problem for torsion-free abelian groups is analytic complete. Journal of Algebra, vol. 320 (2008), no. 6, pp. 2291–2300.CrossRefGoogle Scholar
Eagle, Christopher J., Farah, Ilijas, Hart, Bradd, Kadets, Boris, Kalashnyk, Vladyslav, and Lupini, Martino, Fraïssé limits of C*-algebras, arXiv:1411.4066, 2014.Google Scholar
Effros, Edward G., Some quantizations and reflections inspired by the Gelfand-Naimark theorem, C*-algebras: 1943–1993 (San Antonio, TX, 1993), Contemporary Mathematics, vol. 167, American Mathematical Society, Providence, RI, 1994, pp. 98–113.Google Scholar
Elliott, George A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. Journal of Algebra, vol. 38 (1976), no. 1, pp. 29–44.CrossRefGoogle Scholar
Elliott, George A., Some C*-algebras with outer derivations, III. Annals of Mathematics, vol. 106 (1977), no. 1, pp. 121–143.CrossRefGoogle Scholar
Elliott, George A., The classification problem for amenable C*-algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, 1995, p. 922–932.CrossRefGoogle Scholar
Elliott, George A., Farah, Ilijas, Paulsen, Vern, Rosendal, Christian, Toms, Andrew S., and Törnquist, Asger, The isomorphism relation for separable C*-algebras, Mathematical Research Letters, vol. 20 (2013), no. 6, pp. 1071–1080.CrossRefGoogle Scholar
Elliott, George A. and Toms, Andrew 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.CrossRefGoogle Scholar
Farah, Ilijas, A dichotomy for the Mackey Borel structure, Proceedings of the 11th Asian Logic Conference, World Science Publisher, Hackensack, NJ, 2012, p. 86–93.Google Scholar
Farah, Ilijas, Logic and operator algebras, Proceedings of the International Congress of Mathematicians (Seoul, South Corea), 2014.Google Scholar
Farah, Ilijas, Hart, Bradd, Lupini, Martino, Robert, Leonel, Tikuisis, Aaron P., Vignati, Alessandro, and Winter, Wilhelm, Model theory of nuclear C*-algebras, in preparation.Google Scholar
Farah, Ilijas, Hart, Bradd, and Sherman, David, Model theory of operator algebras II: Model theory. Israel Journal of Mathematics, vol. 201 (2014), no. 1, pp. 477–505.CrossRefGoogle Scholar
Farah, Ilijas, Toms, Andrew S., and Törnquist, Asger, The descriptive set theory of C*-algebra invariants, International Mathematics Research Notices (2012), 5196–5226.Google Scholar
Farah, Ilijas, 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
Ferenczi, Valentin, Louveau, Alain, and Rosendal, Christian, The complexity of classifying separable banach spaces up to isomorphism. Journal of the London Mathematical Society, vol. 79 (2009), no. 2, pp. 323–345.CrossRefGoogle Scholar
Foreman, Matthew and Weiss, Benjamin, An anti-classification theorem for ergodic measure preserving transformations. Journal of the European Mathematical Society, vol. 6 (2004), no. 3, pp. 277–292.CrossRefGoogle Scholar
Friedman, Harvey and Stanley, Lee, A Borel reductibility theory for classes of countable structures, this Journal, vol. 54 (1989), no. 03, pp. 894–914.Google Scholar
Gao, Su, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009.Google Scholar
Gao, Su and Kechris, Alexander S., On the classification of Polish metric spaces up to isometry. Memoirs of the American Mathematical Society, vol. 161 (2003), no. 766, pp. 0–0.CrossRefGoogle Scholar
Gardella, Eusebio and Lupini, Martino, Conjugacy and cocycle conjugacy of automorphisms of ${\cal O}_2 $are not Borel, arXiv:1404.3617, 2014.Google Scholar
Gelfand, Izrail M. and Neumark, Mark A., On the imbedding of normed rings into the ring of operators in Hilbert space. Matematicheskii Sbornik Novaya Seriya, vol. 12 (1943), no. 54, pp. 197–213.Google Scholar
Glimm, James, Type I C*-algebras. Annals of Mathematics, Second Series vol. 73 (1961), pp. 572–612.CrossRefGoogle Scholar
Gong, Guihua, Jiang, Xinhui, and Su, Hongbing, Obstructions to ${\cal Z}$-stability for unital simple C*-algebras, vol. 43 (2000), no. 4, pp. 418–426.Google Scholar
Harrington, Leo A., Kechris, Alexander S., and Louveau, Alain, A Glimm-Effros dichotomy for Borel equivalence relations. Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903–928.CrossRefGoogle Scholar
Harrison, Joseph, Recursive pseudo-well-orderings. Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526–543.CrossRefGoogle Scholar
Hjorth, Greg, Non-smooth infinite-dimensional group representations, preprint, 1997.Google Scholar
Hjorth, Greg, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000.Google Scholar
Hjorth, Greg, On invariants for measure preserving transformations. Fundamenta Mathematicae, vol. 169 (2001), no. 1, pp. 51–84.CrossRefGoogle Scholar
Hjorth, Greg, The isomorphism relation on countable torsion free abelian groups. Fundamenta Mathematicae. vol. 175 (2002), no. 3, pp. 241–257.CrossRefGoogle Scholar
Hjorth, Greg and Kechris, Alexander S., New dichotomies for Borel equivalence relations. The Bulletin of Symbolic Logic, vol. 3 (1997), no. 3, pp. 329–346.CrossRefGoogle Scholar
Izumi, Masaki, Finite group actions on C*-algebras with the Rohlin property, I. Duke Mathematical Journal, vol. 122 (2004), no. 2, pp. 233–280.CrossRefGoogle Scholar
Izumi, Masaki, Finite group actions on C*-algebras with the Rohlin property—II. Advances in Mathematics, vol. 184 (2004), no. 1, pp. 119–160.CrossRefGoogle Scholar
Jiang, Xinhui and Su, Hongbing, On a simple unital projectionless C*-algebra. American Journal of Mathematics, vol. 121 (1999), no. 2, pp. 359–413.CrossRefGoogle Scholar
Kechris, Alexander S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.Google Scholar
Kechris, Alexander S., New directions in descriptive set theory. The Bulletin of Symbolic Logic, vol. 5 (1999), no. 2, pp. 161–174.CrossRefGoogle Scholar
Kechris, Alexander S., Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010.Google Scholar
Kechris, Alexander S. and Sofronidis, Nikolaos E., A strong generic ergodicity property of unitary and self-adjoint operators, Ergodic Theory and Dynamical Systems vol. 21 (2001), no. 5, pp. 1459–1479.CrossRefGoogle Scholar
Kerr, David, Li, Hanfeng, and Pichot, Mikaël, Turbulence, representations, and trace-preserving actions. Proceedings of the London Mathematical Society, vol. 100 (2010), no. 2, pp. 459–484.CrossRefGoogle Scholar
Kerr, David, Lupini, Martino, and Christopher Phillips, N., Borel complexity and automorphisms of C*-algebras. Journal of Functional Analysis, vol. 268 (2015), no. 12, pp. 3767–3789.CrossRefGoogle Scholar
Kirchberg, Eberhard, 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, p. 943–954.Google Scholar
Kirchberg, Eberhard and Christopher Phillips, N., Embedding of exact C*-algebras in the Cuntz algebra ${\cal O}_2 $. Journal für die reine und angewandte Mathematik, vol. 525 (2000), pp. 17–53.CrossRefGoogle Scholar
Kishimoto, Akitaka, The Rohlin property for automorphisms of UHF algebras. Journal für die reine und angewandte Mathematik, vol. 1995 (1995), no. 465, pp. 183–196.Google Scholar
Lindenstrauss, Elon, Mean dimension, small entropy factors and an embedding theorem, Institut des Hautes Études Scientifiques, Publications Mathámatiques (1999), no. 89, 227–262 (2000).CrossRefGoogle Scholar
Lindenstrauss, Elon and Weiss, Benjamin, Mean topological dimension. Israel Journal of Mathematics, vol. 115 (2000), pp. 1–24.CrossRefGoogle Scholar
Louveau, Alain and Rosendal, Christian, Complete analytic equivalence relations. Transactions of the American Mathematical Society, vol. 357 (2005), no. 12, pp. 4839–4866.CrossRefGoogle Scholar
Lupini, Martino, Polish groupoids and functorial complexity, arXiv:1407.6671, 2014.Google Scholar
Lupini, Martino, Unitary equivalence of automorphisms of separable C*-algebras. Advances in Mathematics, 262 (2014), pp. 1002–1034.CrossRefGoogle Scholar
Mackey, George W., Borel structure in groups and their duals. Transactions of the American Mathematical Society, vol. 85 (1957), no. 1, pp. 134–165.CrossRefGoogle Scholar
Melleray, Julien, Computing the complexity of the relation of isometry between separable Banach spaces. Mathematical Logic Quarterly, vol. 53 (2007), no. 2, pp. 128–131.CrossRefGoogle Scholar
Miller, Benjamin D., The graph-theoretic approach to descriptive set theory. Bulletin of Symbolic Logic, vol. 18 (2012), no. 4, pp. 554–575.CrossRefGoogle Scholar
Murray, Francis J. and von Neumann, John, On rings of operators. Annals of Mathematics. vol. 37 (1936), no. 1, pp. 116–229.CrossRefGoogle Scholar
Nakamura, Hideki, Aperiodic automorphisms of nuclear purely infinite simple C*-algebras. Ergodic Theory and Dynamical Systems, vol. 20 (2000), no. 6, pp. 1749–1765.CrossRefGoogle Scholar
Ozawa, Narutaka and Pisier, Gilles, A continuum of C*-norms on $\left( H \right) \otimes \left( H \right)$and related tensor products. Glasgow Mathematical Journal, doi:10.1017/S0017089515000257.CrossRefGoogle Scholar
Phillips, John, Outer automorphisms of separable C*-algebras. Journal of Functional Analysis, vol. 70 (1987), no. 1, pp. 111–116.CrossRefGoogle Scholar
Phillips, John and Raeburn, Iain, Automorphisms of C*-algebras and second Čech cohomology. Indiana University Mathematics Journal, vol. 29 (1980), no. 6, pp. 799–822.CrossRefGoogle Scholar
Christopher Phillips, N., The tracial Rokhlin property is generic, arXiv:1209.3859, 2012.Google Scholar
Popa, Sorin, Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians, Vol. I, European Mathematical Society, Zürich, 2007, pp. 445–477.CrossRefGoogle Scholar
Rørdam, Mikael, A simple C*-algebra with a finite and an infinite projection. Acta Mathematica, vol. 191 (2003), no. 1, pp. 109–142.CrossRefGoogle Scholar
Rørdam, Mikael, The stable and the real rank of ${\cal Z}$-absorbing C*-algebras. International Journal of Mathematics, vol. 15 (2004), no. 10, pp. 1065–1084.CrossRefGoogle Scholar
Rosenberg, Jonathan, Algebraic K-theory and its Applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994.Google Scholar
Rosendal, Christian, The generic isometry and measure preserving homeomorphism are conjugate to their powers. Fundamenta Mathematicae, vol. 205 (2009), no. 1, pp. 1–27.CrossRefGoogle Scholar
Sabok, Marcin, Completeness of the isomorphism problem for separable C*-algebras. Inventiones Mathematicae, 2015, pp. 1–36.Google Scholar
Sato, Yasuhiko, The Rohlin property for automorphisms of the Jiang–Su algebra. Journal of Functional Analysis, vol. 259 (2010), no. 2, pp. 453–476.CrossRefGoogle Scholar
Schochet, Claude, Algebraic topology and C*-algebras, C*-algebras: 1943-1993 (San Antonio, TX, 1993), Contemporary Mathematics, vol. 167, American Mathematical Society, Providence, RI, 1994, pp. 218–231.Google Scholar
Segal, Irving, C*-algebras and quantization, C*-algebraas: 1943–1993 (San Antonio, TX, 1993), Contemporary Mathematics, vol. 167, American Mathematical Society, Providence, RI, 1994, pp. 54–65.Google Scholar
Silver, Jack H., Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Annals of Mathematical Logic, vol. 18 (1980), no. 1, pp. 1–28.CrossRefGoogle Scholar
Toms, Andrew S., On the independence of K-theory and stable rank for simple C*-algebras. Journal für die reine und angewandte Mathematik, vol. 578 (2005), pp. 185–199.Google Scholar
Toms, Andrew S., On the classification problem for nuclear C*-algebras. Annals of Mathematics, vol. 167 (2008), no. 3, pp. 1029–1044.CrossRefGoogle Scholar
Toms, Andrew S., Comparison theory and smooth minimal C*-dynamics. Communications in Mathematical Physics, vol. 289 (2009), no. 2, pp. 401–433.CrossRefGoogle Scholar
Toms, Andrew S. and Winter, Wilhelm, Strongly self-absorbing C*-algebras. Transactions of the American Mathematical Society, vol. 359 (2007), no. 8, pp. 3999–4029.CrossRefGoogle Scholar
Wassermann, Simon, On tensor products of certain group C*-algebras. Journal of Functional Analysis, vol. 23 (1976), no. 3, pp. 239–254.CrossRefGoogle Scholar
Wiersma, Matthew, C*-norms for tensor products of discrete group C*-algebras. Bulletin of the London Mathematical Society, vol. 47 (2015), no. 2, pp. 219–224.CrossRefGoogle Scholar
Zapletal, Jindřich, Forcing Borel reducibility invariants, in preparation.Google Scholar
Zapletal, Jindřich, Analytic equivalence relations and the forcing method. Bulletin of Symbolic Logic, vol. 19 (2013), no. 4, pp. 473–490.CrossRefGoogle Scholar
Zielinski, Joseph, The complexity of the homeomorphism relation between compact metric spaces, arXiv:1409.5523, 2014.Google Scholar