Fell algebras, groupoids, and projections
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- by Robin J. Deeley, Magnus Goffeng and Allan Yashinski PDF
- Proc. Amer. Math. Soc. 150 (2022), 4891-4907 Request permission
Abstract:
Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous-trace $C^*$-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arises naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.References
- R. J. Archbold and D. W. B. Somerset, Transition probabilities and trace functions for $C^*$-algebras, Math. Scand. 73 (1993), no. 1, 81–111. MR 1251700, DOI 10.7146/math.scand.a-12458
- Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645, DOI 10.2140/pjm.1977.71.335
- Lisa Orloff Clark, Astrid an Huef, and Iain Raeburn, The equivalence relations of local homeomorphisms and Fell algebras, New York J. Math. 19 (2013), 367–394. MR 3084709
- Robin J. Deeley, Magnus Goffeng, and Allan Yashinski, Smale space $C^*$-algebras have nonzero projections, Proc. Amer. Math. Soc. 148 (2020), no. 4, 1625–1639. MR 4069199, DOI 10.1090/proc/14837
- Robin J. Deeley and Allan Yashinski, The stable algebra of a Wieler solenoid: inductive limits and $K$-theory, Ergodic Theory Dynam. Systems 40 (2020), no. 10, 2734–2768. MR 4138909, DOI 10.1017/etds.2019.17
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
- A. G. M. Hommelberg, Compact non-Hausdorff manifolds, Bachelor Thesis, Mathematisch Instituut, Universiteit Leiden, June 2014, https://www.math.leidenuniv.nl/scripties/HommelbergBach.pdf.
- Astrid an Huef, Alex Kumjian, and Aidan Sims, A Dixmier-Douady theorem for Fell algebras, J. Funct. Anal. 260 (2011), no. 5, 1543–1581. MR 2749438, DOI 10.1016/j.jfa.2010.11.011
- G. G. Kasparov and G. Skandalis, Groups acting on buildings, operator $K$-theory, and Novikov’s conjecture, $K$-Theory 4 (1991), no. 4, 303–337. MR 1115824, DOI 10.1007/BF00533989
- Ian F. Putnam, $C^*$-algebras from Smale spaces, Canad. J. Math. 48 (1996), no. 1, 175–195. MR 1382481, DOI 10.4153/CJM-1996-008-2
- David Ruelle, Thermodynamic formalism, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. The mathematical structures of equilibrium statistical mechanics. MR 2129258, DOI 10.1017/CBO9780511617546
Additional Information
- Robin J. Deeley
- Affiliation: Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, Colorado 80309-0395
- MR Author ID: 741108
- Email: robin.deeley@colorado.edu
- Magnus Goffeng
- Affiliation: Centre for Mathematical Sciences, University of Lund, Box 118, 221 00 Lund, Sweden
- MR Author ID: 895436
- ORCID: 0000-0002-4411-5104
- Email: magnus.goffeng@math.lth.se
- Allan Yashinski
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
- MR Author ID: 1137217
- Email: ayashins@umd.edu
- Received by editor(s): October 18, 2021
- Received by editor(s) in revised form: February 1, 2022
- Published electronically: July 15, 2022
- Additional Notes: The first author was funded by NSF Grant DMS 2000057 and was previously funded by Simons Foundation Collaboration Grant for Mathematicians number 638449. The second author was supported by the Swedish Research Council Grant VR 2018-0350
- Communicated by: Adrian Ioana
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 4891-4907
- MSC (2020): Primary 46L85, 46L80
- DOI: https://doi.org/10.1090/proc/16045