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Nonclassifiability of UHF \(L^p\)-operator algebras.(English)Zbl 1337.47105

Summary: For \( p\in [1,\infty )\), we prove that simple, separable, monotracial UHF \( L^{p}\)-operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF \( L^{p}\)-operator algebras of tensor product type obtained from a diagonal system of similarities. For \( p=2\), it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of {N. Christopher Phillips}, rely on Borel complexity theory, and particularly Hjorth’s theory of turbulence [G. Hjorth, Classification and orbit equivalence relations. Providence, RI: American Mathematical Society (AMS) (2000;Zbl 0942.03056)].

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
03E15 Descriptive set theory
47L30 Abstract operator algebras on Hilbert spaces
46L80 \(K\)-theory and operator algebras (including cyclic theory)

Citations:

Zbl 0942.03056

Cite

References:

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[3]Daws, Matthew, \(p\)-operator spaces and Fig\`a-Talamanca-Herz algebras, J. Operator Theory, 63, 1, 47-83 (2010) ·Zbl 1199.46125
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[6]Foreman, Matthew; Weiss, Benjamin, An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. (JEMS), 6, 3, 277-292 (2004) ·Zbl 1063.37004
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[8]Gardella, Eusebio; Lupini, Martino, Representations of \'{e}tale groupoids on {\(L^p\)}-spaces, arXiv:1408.3752 (2014) ·Zbl 06769054
[9]Hjorth, Greg, Non-smooth infinite-dimensional group representations (1997)
[10]Hjorth, Greg, Classification and orbit equivalence relations, Mathematical Surveys and Monographs 75, xviii+195 pp. (2000), American Mathematical Society, Providence, RI ·Zbl 0942.03056
[11]Hjorth, Greg, Measuring the classification difficulty of countable torsion-free abelian groups, Rocky Mountain J. Math.. Proceedings of the Second Honolulu Conference on Abelian Groups and Modules (Honolulu, HI, 2001), 32, 4, 1267-1280 (2002) ·Zbl 1040.20045 ·doi:10.1216/rmjm/1181070022
[12]Kerr, David; Li, Hanfeng; Pichot, Mika{\"e}l, Turbulence, representations, and trace-preserving actions, Proc. Lond. Math. Soc. (3), 100, 2, 459-484 (2010) ·Zbl 1192.46063 ·doi:10.1112/plms/pdp036
[13]Kerr, David; Lupini, Martino; Phillips, N. Christopher, Borel complexity and automorphisms of {C}*-algebras, Journal of Functional Analysis ·Zbl 1327.46059
[14]Kechris, A. S.; Sofronidis, N. E., A strong generic ergodicity property of unitary and self-adjoint operators, Ergodic Theory Dynam. Systems, 21, 5, 1459-1479 (2001) ·Zbl 1062.47514 ·doi:10.1017/S0143385701001705
[15]Lupini, Martino, Unitary equivalence of automorphisms of separable \({\rm C}^*\)-algebras, Adv. Math., 262, 1002-1034 (2014) ·Zbl 1291.03088 ·doi:10.1016/j.aim.2014.05.022
[16]Marker, David, Model theory, Graduate Texts in Mathematics 217, viii+342 pp. (2002), Springer-Verlag, New York ·Zbl 1003.03034
[17]Phillips, N. Christopher, Analogs of {C}untz algebras on {\(L^p\)} spaces, arXiv:1201.4196 (2012)
[18]Phillips, N. Christopher, Crossed products of {\(L^p\)} operator algebras and the {K}-theory of {C}untz algebras on {\(L^p\)} spaces, arXiv:1309.6406 (2013)
[19]Phillips, N. Christopher, Isomorphism, nonisomorphism, and amenability of {\(L^p\)} {UHF} algebras, arXiv:1309.3694 (2013)
[20]Phillips, N. Christopher, Simplicity of {UHF} and {C}untz algebras on {\(L^p\)}-spaces, arXiv:1309.0115 (2013)
[21]Phillips, N. Christopher; Viola, Maria Grazia, \(L^p\) analogs of {AF} algebras ·Zbl 07301539
[22]Sasyk, Rom\'{a}n; T{\"{o}}rnquist, Asger, The classification problem for von {N}eumann factors, Journal of Functional Analysis, 256, 8, 2710- 2724 (2009-04) ·Zbl 1173.46042
[23]Sasyk, Rom{\'a}n; T{\"o}rnquist, Asger, Turbulence and {A}raki-{W}oods factors, Journal of Functional Analysis, 259, 9, 2238- 2252 (2010) ·Zbl 1208.46059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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