Abstract
We show that every groupoid C*-algebra is isomorphic to its opposite, and deduce that there exist C*-algebras that are not stably isomorphic to groupoid C*-algebras, though many of them are stably isomorphic to twisted groupoid C*-algebras. We also prove that the opposite algebra of a section algebra of a Fell bundle over a groupoid is isomorphic to the section algebra of a natural opposite bundle.
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L. G. Brown, P. Green and M. A. Rieffel, Stable isomorphism and strong Morita equivalence of C*-algebras, Pacific Journal of Mathematics 71 (1977), 349–363.
A. Connes, A factor not anti-isomorphic to itself, Annals Mathematics 101 (1975), 536–554.
M. Dadarlat, I. Hirshberg and N. C. Phillips, Simple nuclear C*-algebras not equivariantly isomorphic to their opposites, Journal of Noncommutative Geometry 12 (2018), 1227–1253.
R. Exel, Approximately finite C*-algebras and partial automorphisms, Mathematica Scandinavica 77 (1995), 281–288.
R. Exel and E. Pardo Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras, Advances in Mathematics 306 (2017), 1046–1129.
I. Farah and I. Hirshberg, Simple nuclear C*-algebras not isomorphic to their opposites, Proceedings of the National Academy of Sciences of the United States of America 114 (2017), 6244–6249.
E. Gardella and M. Lupini, Representations of étale groupoids in Lp-spaces, Advances in Mathematics 318 (2017), 233–278.
A. an Huef, A. Kumjian and A. Sims, A Dixmier-Douady theorem for Fell algebras, Journal of Functional Analysis 260 (2011), 1543–1581.
A. Kumjian, On C*-diagonals, Canadian Journal of Mathematics 38 (1986), 969–1008.
A. Kumjian, Fell bundles over groupoids, Proceedings of the American Mathematical Society 126 (1998), 1115–1125.
N. C. Phillips, Continuous-trace C*-algebras not isomorphic to their opposite algebras, International Journal of Mathematics 12 (2001), 263–275.
N. C. Phillips, A simple separable C*-algebra not isomorphic to its opposite algebra, Proceedings of the American Mathematical Society 132 (2004), 2997–3005.
N. C. Phillips and M. G. Viola, A simple separable exact C*-algebra not anti-isomorphic to itself, Mathematische Annalen 355 (2013), 783–799.
I. Raeburn and J. L. Taylor, Continuous trace C*-algebras with given Dixmier-Douady class, Journal of the Australian Mathematical Society. Series A 38 (1985), 394–407.
J. Renault, A Groupoid Approach to C*-algebras, Lecture Notes in Mathematics, Vol. 793, Springer, Berlin, 1980.
J. Renault, Cartan subalgebras in C*-algebras, Irish Mathematical Society Bulletin 61 (2008), 29–63.
M. Rørdam and E. Størmer, Classification of nuclear, simple C*-algebras, in Classification of Nuclear C*-algebras. Entropy in Operator Algebras, Encyclopaedia of Mathematical Sciences, Vol. 126, Springer, Berlin, 2002, pp. 1–145.
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The first author is supported by CNPq (Brazil).
The second author was supported by the Australian Research Council grant DP150101598.
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Buss, A., Sims, A. Opposite algebras of groupoid C*-algebras. Isr. J. Math. 244, 759–774 (2021). https://doi.org/10.1007/s11856-021-2190-5
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DOI: https://doi.org/10.1007/s11856-021-2190-5