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Toader, Iulian D.
2023.
Einstein Completeness as Categoricity.
Foundations of Physics,
Vol. 53,
Issue. 2,
Article contents
AN INVITATION TO MODEL THEORY AND C*-ALGEBRAS
Published online by Cambridge University Press: 01 April 2019
Abstract
We present an introductory survey to first order logic for metric structures and its applications to C*-algebras.
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References
REFERENCES
Akemann, C. A. and Pedersen, G. K., Central sequences and inner derivations of separable C*-algebras. American Journal of Mathematics, vol. 101 (1979), no. 5, pp. 1047–1061.CrossRefGoogle Scholar
Arzhantseva, G. and Paunescu, L., Almost commuting permutations are near commuting permutations. Journal of Functional Analysis, vol. 269 (2015), no. 3, pp. 745–757.CrossRefGoogle Scholar
Arzhantseva, G. and Paunescu, L., Linear sofic groups and algebras. Transactions of the American Mathematical Society, vol. 369 (2017), no. 4, pp. 2285–2310.CrossRefGoogle Scholar
Atiyah, M. F. and Hirzebruch, H., Riemann-Roch theorems for differentiable manifolds, Bulletin of the American Mathematical Society. vol. 65 (1959), no. 4, pp. 276–281.CrossRefGoogle Scholar
Bankston, P., Co-elementary equivalence, co-elementary maps, and generalized arcs. Proceedings of the American Mathematical Society, vol. 125 (1977), no. 12, pp. 3715–3720.CrossRefGoogle Scholar
Bankston, P., Reduced coproducts of compact Hausdorff spaces. The Journal of Symbolic Logic, vol. 52 (1987), no. 2, pp. 404–424.CrossRefGoogle Scholar
Bankston, P., A hierarchy of maps between compacta. The Journal of Symbolic Logic, vol. 64 (1999), no. 4, pp. 1628–1644.CrossRefGoogle Scholar
Bankston, P., Some applications of the ultrapower theorem to the theory of compacta. Applied Categorical Structures, vol. 8 (2000 [1997]), no. 1, pp. 45–66.CrossRefGoogle Scholar
Barlak, S. and Szabó, G., Sequentially split *-homomorphisms between C*-algebras. International Journal of Mathematics, vol. 27 (2016), no. 13, p. 1650105.CrossRefGoogle Scholar
Barlak, S., Szabó, G., and Voigt, C., The spatial Rokhlin property for actions of compact quantum groups. Journal of Functional Analysis, vol. 272 (2017), no. 6, pp. 2308–2360.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. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427.CrossRefGoogle Scholar
Ben Yaacov, I. and Iovino, J., Model theoretic forcing in analysis. Annals of Pure and Applied Logic, vol. 158 (2009), no. 3, pp. 163–174.CrossRefGoogle Scholar
Blackadar, B., Encyclopaedia of Mathematical Sciences, Operator Algebras, vol. 122, Springer-Verlag, Berlin, 2006.CrossRefGoogle Scholar
Blackadar, B. and Kirchberg, E., Generalized inductive limits of finite-dimensional C*-algebras. Mathematische Annalen, vol. 307 (1997), no. 3, pp. 343–380.CrossRefGoogle Scholar
Boutonnet, R., Chifan, I., and Ioana, A., II1 factors with nonisomorphic ultrapowers. Duke Mathematical Journal, vol. 166 (2017), no. 11, pp. 2023–2051.CrossRefGoogle Scholar
Brown, L. G., Douglas, R. G., and Fillmore, P. A., Extensions of C*-algebras and K-homology. Annals of Mathematics. Second Series, vol. 105 (1977), no. 2, pp. 265–324.Google Scholar
Brown, L. G. and Pedersen, G., C*-algebras of real rank zero. Journal of Functional Analysis, vol. 99 (1991), no. 1, pp. 131–149.CrossRefGoogle Scholar
Capraro, V. and Lupini, M., Lecture Notes in Mathematics, Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture, vol. 2136, Springer, Cham, 2015.CrossRefGoogle Scholar
Carlson, K., Cheung, E., Farah, I., Gerhardt-Bourke, A., Hart, B., Mezuman, L., Sequeira, N., and Sherman, A., Omitting types and AF algebras. Archive for Mathematical Logic, vol. 53 (2014), no. 1–2, pp. 157–169.CrossRefGoogle Scholar
Carrión, J. R., Dadarlat, M., and Eckhardt, C., On groups with quasidiagonal C*-algebras. Journal of Functional Analysis, vol. 265 (2013), no. 1, pp. 135–152.CrossRefGoogle Scholar
Cohen, P. J., The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, vol. 50 (1963), pp. 1143–1148.CrossRefGoogle ScholarPubMed
Cohen, P. J., The independence of the continuum hypothesis. II. Proceedings of the National Academy of Sciences of the United States of America, vol. 51 (1964), pp. 105–110.CrossRefGoogle ScholarPubMed
Coskey, S. and Farah, I., Automorphisms of corona algebras, and group cohomology. Transactions of the American Mathematical Society, vol. 366 (2014), no. 7, pp. 3611–3630.CrossRefGoogle Scholar
Cuntz, J., Simple C*-algebras generated by isometries. Communications in Mathematical Physics, vol. 57 (1977), no. 2, pp. 173–185.CrossRefGoogle Scholar
Eagle, C. J., Goldbring, I., and Vignati, A., The pseudoarc is a co-existentially closed continuum. Topology and Its Applications, vol. 207 (2016), pp. 1–9.CrossRefGoogle Scholar
Eagle, C. J. and Vignati, A., Saturation and elementary equivalence of C*-algebras. Journal of Functional Analysis, vol. 269 (2015), no. 8, pp. 2631–2664.CrossRefGoogle Scholar
Elek, G. and Szabó, E., Sofic groups and direct finiteness. Journal of Algebra, vol. 280 (2004), no. 2, pp. 426–434.CrossRefGoogle Scholar
Elek, G. and Szabó, E., Sofic representations of amenable groups. Proceedings of the American Mathematical Society, vol. 139 (2011), no. 12, pp. 4285–4291.CrossRefGoogle Scholar
Elliott, G. and Toms, A., 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
Elliott, G. 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, G. A., The classification problem for amenable C*-algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Chatterji, S. D., editor), Birkhäuser, Basel, 1995, pp. 922–932.CrossRefGoogle Scholar
Farah, I., All automorphisms of all Calkin algebras. Mathematical Research Letters, vol. 18 (2011), no. 3, pp. 489–503.CrossRefGoogle Scholar
Farah, I., All automorphisms of the Calkin algebra are inner. Annals of Mathematics, vol. 173 (2011), no. 2, pp. 619–661.CrossRefGoogle Scholar
Farah, I., Goldbring, I., Hart, B., and Sherman, D., Existentially closed II1 factors. Fundamenta Mathematicae, vol. 233 (2016), no. 2, pp. 173–196.Google Scholar
Farah, I. and Hart, B., Countable saturation of corona algebras. Comptes Rendus Mathématiques de l’Académie des Sciences, 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 C*-algebras. Memoirs of the American Mathematical Society, to appear.Google Scholar
Farah, I., Hart, B., Rørdam, M., and Tikuisis, A., Relative commutants of strongly self-absorbing C*-algebras. Selecta Mathematica, vol. 23 (2017), no. 1, pp. 363–387.CrossRefGoogle Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras I: Stability. Bulletin of the London Mathematical Society, vol. 45 (2013), no. 4, pp. 825–838.CrossRefGoogle Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras II: Model theory. Israel Journal of Mathematics, vol. 201 (2014), no. 1, pp. 477–505.CrossRefGoogle Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras III: Elementary equivalence and II1: factors. Bulletin of the London Mathematical Society, vol. 46 (2014), no. 3, pp. 609–628.CrossRefGoogle Scholar
Farah, I. and Hirshberg, I., The Calkin algebra is not countably homogeneous. Proceedings of the American Mathematical Society, vol. 144 (2016), no. 12, pp. 5351–5357.CrossRefGoogle Scholar
Farah, I. and Magidor, M., Omitting types in logic of metric structures, Journal of Mathematical Logic, vol. 18 (2018), no. 2, p. 1850006.CrossRefGoogle Scholar
Farah, I. and McKenney, P., Homeomorphisms of Cech-Stone remainders: The zero-dimensional case, Proccedings of the American Mathematical Society, vol. 146 (2018), no. 5, pp. 2253–2262.CrossRefGoogle Scholar
Farah, I., McKenney, P., and Schimmerling, E., Some Calkin algebras have outer automorphisms. Archive for Mathematical Logic, vol. 52 (2013), no. 5, pp. 517–524.CrossRefGoogle Scholar
Farah, I. and Shelah, S., A dichotomy for the number of ultrapowers. Journal of Mathematical Logic, vol. 10 (2010), no. 1–2, pp. 45–81.CrossRefGoogle Scholar
Farah, I. and Shelah, S., Rigidity of continuous quotients. Journal of the Institute of Mathematics of Jussieu, vol. 15 (2016), no. 1, pp. 1–28.CrossRefGoogle Scholar
Gardella, E., Rokhlin dimension for compact group actions. Indiana University Mathematics Journal, vol. 66 (2017), no. 2, pp. 659–703.CrossRefGoogle Scholar
Gardella, E, Kalantar, M., and Lupini, M., Rokhlin dimension for compact quantum group actions. Journal of Noncommutative Geometry, to appear.Google Scholar
Gardella, E. and Lupini, M., Applications of model theory to C*-dynamics. Journal of Functional Analysis, vol. 275 (2018), no. 7, pp. 1889–1942.CrossRefGoogle Scholar
Glebsky, L., Almost commuting matrices with respect to normalized Hilbert-Schmidt norm, preprint, 2010, arXiv:1002.3082.Google 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.CrossRefGoogle Scholar
Gödel, K., The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, vol. 3, Princeton University Press, Princeton, NJ, 1940.Google Scholar
Goldbring, I., Hart, B., and Sinclair, T., The theory of tracial von Neumann algebras does not have a model companion. Journal of Symbolic Logic, vol. 78 (2013), no. 3, pp. 1000–1004.CrossRefGoogle Scholar
Goldbring, I. and Sinclair, T., Robinson forcing and the quasidiagonality problem. International Journal of Mathematics, vol. 28 (2017), no. 2, p. 1750008.CrossRefGoogle Scholar
Goldbring, I. and Sinclair, T., On Kirchberg’s embedding problem. Journal of Functional Analysis, vol. 269 (2015), no. 1, pp. 155–198.CrossRefGoogle Scholar
Gromov, M., Endomorphisms of symbolic algebraic varieties. Journal of the European Mathematical Society, vol. 1 (1999), no. 2, pp. 109–197.CrossRefGoogle Scholar
Hodges, W., Building Models by Games, London Mathematical Society Student Texts, vol. 2, Cambridge University Press, Cambridge, 1985.Google Scholar
Jiang, X. and Su, H., On a simple unital projectionless C*-algebra. American Journal of Mathematics, vol. 121 (1999), no. 2, pp. 359–413.Google Scholar
Kirchberg, E. and Phillips, N. C., 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
${{\cal O}_2}$.. Journal für die reine und angewandte Mathematik, vol. 525 (2000), pp. 17–53.CrossRefGoogle ScholarKirchberg, E. and Rørdam, M., Infinite non-simple C*-algebras: Absorbing the Cuntz algebra ${{\cal O}_\infty }$.. Advances in Mathematics, vol. 167 (2002), no. 2, pp. 195–264.CrossRefGoogle Scholar
${{\cal O}_\infty }$.. Advances in Mathematics, vol. 167 (2002), no. 2, pp. 195–264.CrossRefGoogle ScholarLoring, T. A., Fields Institute Monographs, Lifting Solutions to Perturbing Problems in C*-Algebras, vol. 8, American Mathematical Society, Providence, RI, 1997.Google Scholar
McDuff, D., Central sequences and the hyperfinite factor. Proceedings of the London Mathematical Society. Third Series, vol. 21 (1970), pp. 443–461.Google Scholar
Ozawa, N., About the QWEP conjecture. International Journal of Mathematics, vol. 15 (2004), no. 5, pp. 501–530.CrossRefGoogle Scholar
Pedersen, G. K., Graduate Texts in Mathematics, Analysis Now, vol. 118, Springer-Verlag, New York, 1989.CrossRefGoogle Scholar
Pestov, V. G., Hyperlinear and sofic groups: A brief guide, this Journal, vol. 14 (2008), no. 4, pp. 449–480.Google Scholar
Phillips, N. C., Real rank and exponential length of tensor products with ${{\cal O}_\infty }$.. Journal of Operator Theory, vol. 47 (2002), no. 1, pp. 117–130.Google Scholar
${{\cal O}_\infty }$.. Journal of Operator Theory, vol. 47 (2002), no. 1, pp. 117–130.Google ScholarPhillips, N. C. and Weaver, N., The Calkin algebra has outer automorphisms. Duke Mathematical Journal, vol. 139 (2007), no. 1, pp. 185–202.CrossRefGoogle Scholar
Podleś, P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU2 and SO3 groups. Communications in Mathematical Physics, vol. 170 (1995), no. 1, pp. 1–20.CrossRefGoogle Scholar
Raeburn, I. and Williams, D. P., Mathematical Surveys and Monographs, Morita Equivalence and Continuous-Trace C*-Algebras, vol. 60, American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
Rørdam, M., Encyclopaedia of Mathematical Sciences, Classification of Nuclear C*-Algebras, vol. 126, Springer-Verlag, Berlin, 2002.CrossRefGoogle Scholar
Rosenberg, J., Graduate Texts in Mathematics, Algebraic K-Theory and Its Applications, vol. 147, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
Schochet, C., Algebraic topology and C*-algebras, C*-Algebras: 1943–1993 (Doran, R. S., editor), Contemporary Mathematics, vol. 167, American Mathematical Society, Providence, RI, 1994, pp. 218–231.CrossRefGoogle Scholar
Shoenfield, J. R., The problem of predicativity, Essays on the Foundations of Mathematics (Bar-Hillel, Y., Poznanski, E. I. J., Rabin, M. O., and Robinson, A., editors), Magnes Press, Hebrew University of Jerusalem, 1961, pp. 132–139.Google Scholar
Szabó, G., Strongly self-absorbing C*-dynamical systems. Transactions of the American Mathematical Society, vol. 370 (2018), no. 1, pp. 99–130.CrossRefGoogle Scholar
Szabó, G., Strongly self-absorbing C*-dynamical systems, II. Journal of Noncommutative Geometry, vol. 12 (2018), no. 1, pp. 369–406.CrossRefGoogle Scholar
Szabó, G., Strongly self-absorbing C*-dynamical systems, III. Advances in Mathematics, vol. 316 (2017), pp. 356–380.CrossRefGoogle Scholar
Thom, A., Sofic groups and diophantine approximation. Communications on Pure and Applied Mathematics, vol. 61 (2008), no. 8, pp. 1155–1171.CrossRefGoogle Scholar
Thom, A., About the metric approximation of Higman’s group. Journal of Group Theory, vol. 15 (2012), no. 2, pp. 301–310.CrossRefGoogle Scholar
Todorcevic, S., Annals of Mathematics Studies, Introduction to Ramsey Spaces, vol. 174, Princeton University Press, Princeton, NJ, 2010.Google Scholar
Toms, A., An infinite family of non-isomorphic C*-algebras with identical K-theory. Transactions of the American Mathematical Society, vol. 360 (2008), no. 10, pp. 5343–5354.CrossRefGoogle Scholar
Toms, A. S., Comparison theory and smooth minimal C*-dynamics. Communications in Mathematical Physics, vol. 289 (2009), no. 2, pp. 401–433.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
Vignati, A., Nontrivial homeomorphisms of Cech-Stone remainders. Münster Journal of Mathematics, vol. 10 (2017), no. 1, pp. 189–200.Google Scholar
Voiculescu, D., Asymptotically commuting finite rank unitary operators without commuting approximants. Acta Universitatis Szegediensis, vol. 45 (1983), no. 1–4, pp. 429–431.Google Scholar
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