Intermediate C*-algebras of Cartan embeddings

Brown, Jonathan; Exel, Ruy; Fuller, Adam; Pitts, David; Reznikoff, Sarah

doi:10.1090/bproc/66

Intermediate C$^*$-algebras of Cartan embeddings
HTML articles powered by AMS MathViewer

by Jonathan H. Brown, Ruy Exel, Adam H. Fuller, David R. Pitts and Sarah A. Reznikoff HTML | PDF

Proc. Amer. Math. Soc. Ser. B 8 (2021), 27-41

Abstract:

Let $A$ be a C$^*$-algebra and let $D$ be a Cartan subalgebra of $A$. We study the following question: if $B$ is a C$^*$-algebra such that $D \subseteq B \subseteq A$, is $D$ a Cartan subalgebra of $B$? We give a positive answer in two cases: the case when there is a faithful conditional expectation from $A$ onto $B$, and the case when $A$ is nuclear and $D$ is a C$^*$-diagonal of $A$. In both cases there is a one-to-one correspondence between the intermediate C$^*$-algebras $B$, and a class of open subgroupoids of the groupoid $G$, where $\Sigma \rightarrow G$ is the twist associated with the embedding $D \subseteq A$.

References

C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683
Hisashi Aoi, A construction of equivalence subrelations for intermediate subalgebras, J. Math. Soc. Japan 55 (2003), no. 3, 713–725. MR 1978219, DOI 10.2969/jmsj/1191418999
R. J. Archbold, J. W. Bunce, and K. D. Gregson, Extensions of states of $C^{\ast }$-algebras. II, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 1-2, 113–122. MR 667130, DOI 10.1017/S0308210500019983
Erik Bédos and Roberto Conti, Fourier series and twisted $\textrm {C}^{\ast }$-crossed products, J. Fourier Anal. Appl. 21 (2015), no. 1, 32–75. MR 3302101, DOI 10.1007/s00041-014-9360-3
Jonathan H. Brown, Adam H. Fuller, David R. Pitts, and Sarah A. Reznikoff, Graded C*-algebras and twisted groupoid C*-algebras, arXiv e-prints (2019), arXiv:1909.04710.
Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, and Dana P. Williams, Cartan subalgebras in $C^*$-algebras of Hausdorff étale groupoids, Integral Equations Operator Theory 85 (2016), no. 1, 109–126. MR 3503181, DOI 10.1007/s00020-016-2285-2
Nathanial P. Brown and Narutaka Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387, DOI 10.1090/gsm/088
Jan Cameron, David R. Pitts, and Vrej Zarikian, Bimodules over Cartan MASAs in von Neumann algebras, norming algebras, and Mercer’s theorem, New York J. Math. 19 (2013), 455–486. MR 3104558
Jan Cameron and Roger R. Smith, Bimodules in crossed products of von Neumann algebras, Adv. Math. 274 (2015), 539–561. MR 3318160, DOI 10.1016/j.aim.2014.12.038
J. Cameron and R. R. Smith, A Galois correspondence for reduced crossed products of simple $\text {C}^{\ast }$ -algebras by discrete groups, Canadian Journal of Mathematics (2019), 1–23.
Hisashi Choda, A Galois correspondence in a von Neumann algebra, Tohoku Math. J. (2) 30 (1978), no. 4, 491–504. MR 516882, DOI 10.2748/tmj/1178229909
Hisashi Choda, A correspondence between subgroups and subalgebras in a discrete $C^{\ast }$-crossed product, Math. Japon. 24 (1979/80), no. 2, 225–229. MR 550000
Michael Cowling and Uffe Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549. MR 996553, DOI 10.1007/BF01393695
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
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
A. P. Donsig, A. H. Fuller, and D. R. Pitts, Cartan Triples, arXiv e-prints (2018), arXiv:1810.05267.
Allan P. Donsig and David R. Pitts, Coordinate systems and bounded isomorphisms, J. Operator Theory 59 (2008), no. 2, 359–416. MR 2411050
Ruy Exel, Amenability for Fell bundles, J. Reine Angew. Math. 492 (1997), 41–73. MR 1488064, DOI 10.1515/crll.1997.492.41
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
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
Igor Fulman, Crossed products of von Neumann algebras by equivalence relations and their subalgebras, Mem. Amer. Math. Soc. 126 (1997), no. 602, x+107. MR 1371091, DOI 10.1090/memo/0602
Thierry Giordano, David Kerr, N. Christopher Phillips, and Andrew Toms, Crossed products of $C^*$-algebras, topological dynamics, and classification, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Cham, 2018. Lecture notes based on the course held at the Centre de Recerca Matemàtica (CRM) Barcelona, June 14–23, 2011; Edited by Francesc Perera. MR 3837150, DOI 10.1007/978-3-319-70869-0
Masaki Izumi, Inclusions of simple $C^\ast$-algebras, J. Reine Angew. Math. 547 (2002), 97–138. MR 1900138, DOI 10.1515/crll.2002.055
Masaki Izumi, Roberto Longo, and Sorin Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998), no. 1, 25–63. MR 1622812, DOI 10.1006/jfan.1997.3228
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 and Ralf Meyer, Noncommutative Cartan $\rm C^*$-subalgebras, Trans. Amer. Math. Soc. 373 (2020), no. 12, 8697–8724. MR 4177273, DOI 10.1090/tran/8174
Richard Mercer, Convergence of Fourier series in discrete crossed products of von Neumann algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 254–258. MR 784174, DOI 10.1090/S0002-9939-1985-0784174-0
Soyoung Moon, Amenable actions of discrete groups, Ph.D. thesis, Universitè de Neuchâtel, 2009.
Paul S. Muhly, Chao Xin Qiu, and Baruch Solel, Coordinates, nuclearity and spectral subspaces in operator algebras, J. Operator Theory 26 (1991), no. 2, 313–332. MR 1225519
David R. Pitts, Structure for regular inclusions. I, J. Operator Theory 78 (2017), no. 2, 357–416. MR 3725511, DOI 10.7900/jot
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017
Aidan Sims and Dana P. Williams, Amenability for Fell bundles over groupoids, Illinois J. Math. 57 (2013), no. 2, 429–444. MR 3263040
Takuya Takeishi, On nuclearity of $C^*$-algebras of Fell bundles over étale groupoids, Publ. Res. Inst. Math. Sci. 50 (2014), no. 2, 251–268. MR 3223473, DOI 10.4171/PRIMS/132
M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. MR 1873025
Jun Tomiyama, The interplay between topological dynamics and theory of $C^*$-algebras, Lecture Notes Series, vol. 2, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1992. MR 1160781
G. Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239 (French). MR 241994