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INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

Part of: Infinite-dimensional manifolds Selfadjoint operator algebras

Published online by Cambridge University Press:  27 March 2020

KONRAD AGUILAR
KONRAD AGUILAR*
Affiliation:
Department of Mathematics and Computer Science (IMADA), University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark e-mail: aguilar@imada.sdu.dk
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Abstract

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

Keywords

noncommutative metric geometryMonge–Kantorovich distancequantum metric spacesLip-normsinductive limitsAF algebras

MSC classification

Primary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 46L30: States 58B34: Noncommutative geometry (à la Connes)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 289 - 312
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by L. O. Clark

We gratefully acknowledge the financial support from the Independent Research Fund Denmark through the ‘Classical and Quantum Distances’ project (Grant No. 9040-00107B).

References

Aguilar, K., ‘AF algebras in the quantum Gromov–Hausdorff propinquity space’, submitted, 2016, arXiv:1612.02404.10.4064/sm8478-2-2016CrossRefGoogle Scholar
Aguilar, K., ‘Fell topologies for AF-algebras and the quantum propinquity’, J. Operator Theory 82(2) (2019), 469–514.Google Scholar
Aguilar, K. and Kaad, J., ‘The Podleś sphere as a spectral metric space’, J. Geom. Phys. 133 (2018), 260–278.10.1016/j.geomphys.2018.07.015CrossRefGoogle Scholar
Aguilar, K. and Latrémolière, F., ‘Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity’, Stud. Math. 231(2) (2015), 149–193.Google Scholar
Aguilar, K. and Latrémolière, F., ‘Some applications of conditional expectations to convergence for the quantum Gromov–Hausdorff propinquity’, submitted, 2018, arXiv:1708.00595.Google Scholar
Antonescu, C. and Christensen, E., ‘Spectral triples for AF C ∗ -algebras and metrics on the Cantor set’, J. Operator Theory 56(1) (2006), 17–46.Google Scholar
Boca, F., ‘An algebra associated with the Farey tessellation’, Canad. J. Math. 60(5) (2008), 975–1000.10.4153/CJM-2008-043-1CrossRefGoogle Scholar
Brown, N. P. and Ozawa, N., C ∗ -Algebras and Finite-Dimensional Approximations, Graudate Studies in Mathematics, 88 (American Mathematical Society, Providence, RI, 2008).10.1090/gsm/088CrossRefGoogle Scholar
Burago, D., Burago, Y. and Ivanov, S., A Course in Metric Geometry, Graduate Texts in Mathematics, 33 (American Mathematical Society, Providence, RI, 2001).10.1090/gsm/033CrossRefGoogle Scholar
Christ, M. and Rieffel, M. A., ‘Nilpotent group C∗ -algebras as compact quantum metric spaces’, Canad. Math. Bull. 60(1) (2017), 77–94.10.4153/CMB-2016-040-6CrossRefGoogle Scholar
Connes, A., ‘Compact metric spaces, Fredholm modules and hyperfiniteness’, Ergod. Th. Dynam. Sys. 9(2) (1989), 207–220.10.1017/S0143385700004934CrossRefGoogle Scholar
Connes, A., Noncommutative Geometry (Academic Press, San Diego, 1994).Google Scholar
Dabrowski, L. and Sitarz, A., ‘Dirac operator on the standard Podleś quantum sphere’, in: Noncommutative Geometry and Quantum Groups (Warsaw, 2001), Banach Center Publ., 61 (Polish Academy of Sciences Institute of Mathematics, Warsaw, 2003), 49–58.10.4064/bc61-0-4CrossRefGoogle Scholar
Davidson, K. R., C ∗ -Algebras by Example, Fields Institute Monographs (American Mathematical Society, Providence, RI, 1996).10.1090/fim/006CrossRefGoogle Scholar
Effros, E. G. and Shen, C. L., ‘Approximately finite C ∗ -algebras and continued fractions’, Indiana Univ. Math. J. 29(2) (1980), 191–204.10.1512/iumj.1980.29.29013CrossRefGoogle Scholar
Elliott, G. A., ‘On the classification of inductive limits of sequences of semisimple finite-dimensional algebras’, J. Algebra 38(1) (1976), 29–44.10.1016/0021-8693(76)90242-8CrossRefGoogle Scholar
Fell, J. M. G., ‘The structure of algebras of operator fields’, Acta. Math. 106 (1961), 233–280.10.1007/BF02545788CrossRefGoogle Scholar
Fell, J. M. G., ‘A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space’, Proc. Amer. Math. Soc. 13 (1962), 472–476.10.1090/S0002-9939-1962-0139135-6CrossRefGoogle Scholar
Glimm, J., ‘On a certain class of operator algebras’, Trans. Amer. Math. Soc. 95 (1960), 318–340.10.1090/S0002-9947-1960-0112057-5CrossRefGoogle Scholar
Latrémolière, F., ‘Convergence of fuzzy tori and quantum tori for the quantum Gromov–Hausdorff propinquity: an explicit approach’, Münster J. Math. 8(1) (2015).Google Scholar
Latrémolière, F., ‘The dual Gromov–Hausdorff propinquity’, J. Math. Pure Appl. 103(2) (2015), 303–351.10.1016/j.matpur.2014.04.006CrossRefGoogle Scholar
Latrémolière, F., ‘The quantum Gromov–Hausdorff propinquity’, Trans. Amer. Math. Soc. 368 (2016), 365–4111.10.1090/tran/6334CrossRefGoogle Scholar
Latrémolière, F., ‘Quantum metric spaces and the Gromov–Hausdorff propinquity’, in: Noncommutative Geometry and Optimal Transport, Contemporary Mathematics, 676 (American Mathematical Society, Providence, RI, 2016), 47–133.10.1090/conm/676/13608CrossRefGoogle Scholar
Latrémolière, F., ‘The modular Gromov–Hausdorff propinquity’, submitted, 2016, arXiv:1608.04881.Google Scholar
Latrémolière, F., ‘A compactness theorem for the dual Gromov–Hausdorff propinquity’, Indiana Univ. Math. J. 66(5) (2017), 1707–1753.10.1512/iumj.2017.66.6151CrossRefGoogle Scholar
Latrémolière, F., ‘Semigroupoid, groupoid and group actions on limits for the Gromov–Hausdorff propinquity’, submitted, 2017, arXiv:1708.01973.Google Scholar
Latrémolière, F., ‘Convergence of Cauchy sequences for the covariant Gromov–Hausdorff propinquity’, submitted, 2018, arXiv:1806.04721.10.1016/j.jmaa.2018.09.018CrossRefGoogle Scholar
Latrémolière, F., ‘The covariant Gromov–Hausdorff propinquity’, submitted, 2018, arXiv:1805.11229.10.4064/dm778-5-2019CrossRefGoogle Scholar
Latrémolière, F. and Packer, J., ‘Noncommutative solenoids and the Gromov–Hausdorff propinquity’, Proc. Amer. Math. Soc. 145(5) (2017), 2043–2057.10.1090/proc/13229CrossRefGoogle Scholar
Mundici, D., ‘Farey stellar subdivisions, ultrasimplicial groups, and K 0 of AF C ∗ -algebras’, Adv. Math. 68(1) (1988), 23–39.10.1016/0001-8708(88)90006-0CrossRefGoogle Scholar
Murphy, G. J., C ∗ -Algebras and Operator Theory (Academic Press, San Diego, 1990).Google Scholar
Ozawa, N. and Rieffel, M. A., ‘Hyperbolic group C ∗ -algebras and free products C ∗ -algebras as compact quantum metric spaces’, Canad. J. Math. 57 (2005), 1056–1079.10.4153/CJM-2005-040-0CrossRefGoogle Scholar
Rieffel, M. A., ‘Metrics on states from actions of compact groups’, Doc. Math. 3 (1998), 215–229.Google Scholar
Rieffel, M. A., ‘Metrics on state spaces’, Doc. Math. 4 (1999), 559–600.Google Scholar
Rieffel, M. A., ‘Group C ∗ -algebras as compact quantum metric spaces’, Doc. Math. 7 (2002), 605–651.Google Scholar
Rieffel, M. A., ‘Compact quantum metric spaces’, in: Operator Algebras, Quantization, and Noncommutative Geometry, Contemporary Mathematics, 365 (American Mathematical Society, 2005), 315–330.Google Scholar
Rieffel, M. A., ‘Vector bundles and Gromov–Hausdorff distance’, J. K-theory 5 (2010), 39–103.10.1017/is008008014jkt080CrossRefGoogle Scholar
Rieffel, M. A., ‘Matricial bridges for “Matrix algebras converge to the sphere”’, in: Operator Algebras and their Applications, Contemporary Mathematics, 671 (American Mathematical Society, Providence, RI, 2016), 209–233.10.1090/conm/671/13512CrossRefGoogle Scholar
Rieffel, M. A., ‘Vector bundles for “Matrix algebras converge to the sphere”’, J. Geom. Phys. 132 (2018), 181–204.10.1016/j.geomphys.2018.06.003CrossRefGoogle Scholar
Rieffel, M. A., ‘Gromov–Hausdorff distance for quantum metric spaces’, Mem. Amer. Math. Soc. 168(796) (2004).Google Scholar
Rørdam, M., Larsen, F. and Laustsen, N. J., An Introduction to K-Theory for C ∗ -Algebras, London Mathematical Society Student Texts, 49 (Cambridge University Press, Cambridge, 2000).10.1017/CBO9780511623806CrossRefGoogle Scholar
Willard, S., General Topology (Dover Publications, Mineola, NY, 2004).Google Scholar