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COMPACT METRIZABLE STRUCTURES AND CLASSIFICATION PROBLEMS

Published online by Cambridge University Press:  01 May 2018

CHRISTIAN ROSENDAL  and
JOSEPH ZIELINSKI
CHRISTIAN ROSENDAL
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE (M/C 249) UNIVERSITY OF ILLINOIS AT CHICAGO 851 S. MORGAN ST. CHICAGO, IL60607-7045, USAE-mail:[email protected]: http://homepages.math.uic.edu/∼rosendal
JOSEPH ZIELINSKI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY WEAN HALL 6113 PITTSBURGH, PA15213, USAE-mail:[email protected]: http://math.cmu.edu/∼zielinski/
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Abstract

We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism of C*-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness of the isomorphism relation of separable, simple, nuclear C*-algebras recently established by M. Sabok.

Keywords

03E15Borel reducibilitycompact metrizable structurescomplexity of classification problems
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 165 - 186
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Ahlbrandt, G. and Ziegler, M., Quasi finitely axiomatizable totally categorical theories. Annals of Pure and Applied Logic, vol. 30 (1986), no. 1, pp. 63–82.CrossRefGoogle Scholar
Baer, R., Zur Einführung des Scharbegriffs. Journal für die reine und angewandte Mathematik, vol. 160 (1929), pp. 199–207.CrossRefGoogle Scholar
Becker, H. and Kechris, A. S., The Descriptive Set Theory of Polish Group Actions, vol. 232, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
Certaine, J., The ternary operation (abc) = ab-1c of a group. Bulletin of the American Mathematical Society, vol. 49 (1943), no. 12, pp. 869–877.CrossRefGoogle Scholar
Elliott, G. A., Farah, I., Paulsen, V., Rosendal, C., Toms, A. S., and Törnquist, A., The isomorphism relation for separable C*-algebras. Mathematical Research Letters, vol. 20 (2013), no. 6, pp. 1071–1080.CrossRefGoogle Scholar
Ferenczi, V., Louveau, A., and Rosendal, C., 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
Gao, S. and Kechris, A. S., On the Classification of Polish Metric Spaces up to Isometry, Memoirs of the American Mathematical Society, vol. 766, American Mathematical Society, Providence, RI, 2003.Google Scholar
Gromov, M., Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Boston, 1999.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Kechris, A. S., Nies, A., and Tent, K., The complexity of topological group isomorphism, 2017. Available at https://arxiv.org/pdf/1705.08081.pdf.Google Scholar
Mandelkern, M., Metrization of the one-point compactification. Proceedings of the American Mathematical Society, vol. 107 (1989), no. 4, pp. 1111–1115.CrossRefGoogle Scholar
Roelcke, W. and Dierolf, S., Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill, New York, 1981.Google Scholar
Rosendal, C., Cofinal families of Borel equivalence relations and quasiorders, this Journal, vol. 70 (2005), no. 04, pp. 1325–1340.Google Scholar
Rosendal, C., A topological version of the Bergman property. Forum Mathematicum, vol. 21 (2009), no. 2, pp. 299–332.CrossRefGoogle Scholar
Sabok, M., Completeness of the isomorphism problem for separable C*-algebras. Inventiones Mathematicae, vol. 204 (2016), no. 3, pp. 833–868.CrossRefGoogle Scholar
Solecki, S., Extending partial isometries. Israel Journal of Mathematics, vol. 150 (2005), no. 1, pp. 315–331.CrossRefGoogle Scholar
Uspenskij, V. V., On subgroups of minimal topological groups. Topology and its Applications, vol. 155 (2008), no. 14, pp. 1580–1606.CrossRefGoogle Scholar
van Mill, J, The Infinite-Dimensional Topology of Function Spaces, vol. 64, North-Holland, Amsterdam, 2001.Google Scholar
Zielinski, J., The complexity of the homeomorphism relation between compact metric spaces. Advances in Mathematics, vol. 291 (2016), pp. 635–645.CrossRefGoogle Scholar