Noncommutative Cartan 𝐶*-subalgebras

Kwaśniewski, Bartosz; Meyer, Ralf

doi:10.1090/tran/8174

Noncommutative Cartan $\mathrm {C}^*$-subalgebras
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Trans. Amer. Math. Soc. 373 (2020), 8697-8724 Request permission

Abstract:

We characterise Exel’s noncommutative Cartan subalgebras in several ways using uniqueness of conditional expectations, relative commutants, or purely outer inverse semigroup actions. We describe in which sense the crossed product decomposition for a noncommutative Cartan subalgebra is unique. We relate the property of being a noncommutative Cartan subalgebra to aperiodic inclusions and effectivity of dual groupoids. In particular, we extend Renault’s characterisation of commutative Cartan subalgebras.

References

Selçuk Barlak and Xin Li, Cartan subalgebras and the UCT problem, Adv. Math. 316 (2017), 748–769. MR 3672919, DOI 10.1016/j.aim.2017.06.024
Selçuk Barlak and Xin Li, Cartan subalgebras and the UCT problem, II, Math. Ann. 378 (2020), no. 1-2, 255–287. MR 4150918, DOI 10.1007/s00208-020-02022-4
Lawrence G. Brown, James A. Mingo, and Nien-Tsu Shen, Quasi-multipliers and embeddings of Hilbert $C^\ast$-bimodules, Canad. J. Math. 46 (1994), no. 6, 1150–1174. MR 1304338, DOI 10.4153/CJM-1994-065-5
Nathan Brownlowe and Iain Raeburn, Exel’s crossed product and relative Cuntz-Pimsner algebras, Math. Proc. Cambridge Philos. Soc. 141 (2006), no. 3, 497–508. MR 2281412, DOI 10.1017/S030500410600956X
Alcides Buss and Ruy Exel, Fell bundles over inverse semigroups and twisted étale groupoids, J. Operator Theory 67 (2012), no. 1, 153–205. MR 2881538
Alcides Buss, Ruy Exel, and Ralf Meyer, Reduced $\rm C^*$-algebras of Fell bundles over inverse semigroups, Israel J. Math. 220 (2017), no. 1, 225–274. MR 3666825, DOI 10.1007/s11856-017-1516-9
Alcides Buss and Ralf Meyer, Inverse semigroup actions on groupoids, Rocky Mountain J. Math. 47 (2017), no. 1, 53–159. MR 3619758, DOI 10.1216/RMJ-2017-47-1-53
Toke Meier Carlsen, Efren Ruiz, Aidan Sims, and Mark Tomforde, Reconstruction of groupoids and $C^*$-rigidity of dynamical systems (2017), eprint. arXiv:1711.01052.
Man Duen Choi, A Schwarz inequality for positive linear maps on $C^{\ast } \$-algebras, Illinois J. Math. 18 (1974), 565–574. MR 355615
Lisa Orloff Clark, Ruy Exel, Enrique Pardo, Aidan Sims, and Charles Starling, Simplicity of algebras associated to non-Hausdorff groupoids, Trans. Amer. Math. Soc. 372 (2019), no. 5, 3669–3712. MR 3988622, DOI 10.1090/tran/7840
Allan P. Donsig, Adam H. Fuller, and David R. Pitts, Von Neumann algebras and extensions of inverse semigroups, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 1, 57–97. MR 3589841, DOI 10.1017/S0013091516000183
Ruy Exel, Amenability for Fell bundles, J. Reine Angew. Math. 492 (1997), 41–73. MR 1488064, DOI 10.1515/crll.1997.492.41
Ruy Exel, Inverse semigroups and combinatorial $C^\ast$-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313. MR 2419901, DOI 10.1007/s00574-008-0080-7
Ruy Exel, Noncommutative Cartan subalgebras of $C^*$-algebras, New York J. Math. 17 (2011), 331–382. MR 2811068
Ruy Exel, Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, vol. 224, American Mathematical Society, Providence, RI, 2017. MR 3699795, DOI 10.1090/surv/224
Ruy Exel and David R. Pitts, Characterizing groupoid $\mathrm {C}^*$-algebras of non-Hausdorff étale groupoids (2019), eprint. arXiv:1901.09683.
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 325–359. MR 578730, DOI 10.1090/S0002-9947-1977-0578730-2
Akitaka Kishimoto, Outer automorphisms and reduced crossed products of simple $C^{\ast }$-algebras, Comm. Math. Phys. 81 (1981), no. 3, 429–435. MR 634163
Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0
Bartosz Kosma Kwaśniewski, Exel’s crossed product and crossed products by completely positive maps, Houston J. Math. 43 (2017), no. 2, 509–567. MR 3690127
Bartosz Kosma Kwaśniewski and Ralf Meyer, Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles, Studia Math. 241 (2018), no. 3, 257–303. MR 3756105, DOI 10.4064/sm8762-5-2017
Bartosz Kosma Kwaśniewski and Ralf Meyer, Stone duality and quasi-orbit spaces for generalised $\textrm {C}^\ast$-inclusions, Proc. Lond. Math. Soc. (3) 121 (2020), no. 4, 788–827. MR 4105787, DOI 10.1112/plms.12332
Bartosz Kosma Kwaśniewski and Ralf Meyer, Essential crossed products by inverse semigroup actions: simplicity and pure infiniteness (2019), eprint. arXiv:1906.06202.
E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
Xin Li and Jean Renault, Cartan subalgebras in $\textrm {C}^*$-algebras. Existence and uniqueness, Trans. Amer. Math. Soc. 372 (2019), no. 3, 1985–2010. MR 3976582, DOI 10.1090/tran/7654
Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017
Vrej Zarikian, Unique expectations for discrete crossed products, Ann. Funct. Anal. 10 (2019), no. 1, 60–71. MR 3899956, DOI 10.1215/20088752-2018-0008