Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Cycline subalgebras of $ k$-graph C*-algebras

Author: Dilian Yang
Journal: Proc. Amer. Math. Soc. 144 (2016), 2959-2969
MSC (2010): Primary 46L05
Published electronically: November 4, 2015
MathSciNet review: 3487228
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that the cycline subalgbra of a $ k$-graph C*-algebra is maximal abelian, and show when it is a Cartan subalgebra (in the sense of Renault).

References [Enhancements On Off] (What's this?)

  • [Bla06] B. Blackadar, Operator algebras. Theory of $ C^*$-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, 122; Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin, 2006. MR 2188261 (2006k:46082)
  • [BCS14] J. Brown, L. Clark, and A. Sierakowski, Purely infinite C*-algebras associated to étale groupoids, to appear in Ergodic Theory Dynam. Systems.
  • [BNR14] Jonathan H. Brown, Gabriel Nagy, and Sarah Reznikoff, A generalized Cuntz-Krieger uniqueness theorem for higher-rank graphs, J. Funct. Anal. 266 (2014), no. 4, 2590-2609. MR 3150172,
  • [BNRSW15] J. H. Brown, G. Nagy S. Reznikoff, A. Sims, and D. P. Williams, Cartan subalgebras in C*-algebras of étale Hausdorff groupoids, preprint 2015, available at
  • [CKSS14] Toke Meier Carlsen, Sooran Kang, Jacob Shotwell, and Aidan Sims, The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources, J. Funct. Anal. 266 (2014), no. 4, 2570-2589. MR 3150171,
  • [DPY08] Kenneth R. Davidson, Stephen C. Power, and Dilian Yang, Atomic representations of rank 2 graph algebras, J. Funct. Anal. 255 (2008), no. 4, 819-853. MR 2433954 (2009j:47154),
  • [DY09a] Kenneth R. Davidson and Dilian Yang, Periodicity in rank 2 graph algebras, Canad. J. Math. 61 (2009), no. 6, 1239-1261. MR 2588421 (2010h:46077),
  • [DY09b] Kenneth R. Davidson and Dilian Yang, Representations of higher rank graph algebras, New York J. Math. 15 (2009), 169-198. MR 2511133 (2010e:47178)
  • [ES12] D. Gwion Evans and Aidan Sims, When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?, J. Funct. Anal. 263 (2012), no. 1, 183-215. MR 2920846,
  • [Hop05] Alan Hopenwasser, The spectral theorem for bimodules in higher rank graph $ C^*$-algebras, Illinois J. Math. 49 (2005), no. 3, 993-1000 (electronic). MR 2210272 (2006j:47128)
  • [HPP05] Alan Hopenwasser, Justin R. Peters, and Stephen C. Power, Subalgebras of graph $ C^*$-algebras, New York J. Math. 11 (2005), 351-386 (electronic). MR 2188247 (2007b:47189)
  • [KP00] Alex Kumjian and David Pask, Higher rank graph $ C^\ast $-algebras, New York J. Math. 6 (2000), 1-20. MR 1745529 (2001b:46102)
  • [NR12] Gabriel Nagy and Sarah Reznikoff, Abelian core of graph algebras, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 889-908. MR 2927813,
  • [Rae05] Iain Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR 2135030 (2005k:46141)
  • [Ren80] Jean Renault, A groupoid approach to $ C^{\ast } $-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266 (82h:46075)
  • [Ren08] Jean Renault, Cartan subalgebras in $ C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29-63. MR 2460017 (2009k:46135)
  • [RS07] David I. Robertson and Aidan Sims, Simplicity of $ C^\ast $-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), no. 2, 337-344. MR 2323468 (2008g:46099),
  • [RS99] Guyan Robertson and Tim Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115-144. MR 1713322 (2000j:46109),
  • [Taylo] M.E. Taylor, Fourier analysis and the FFT, lecture notes.
  • [Tho10] Klaus Thomsen, Semi-étale groupoids and applications, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 3, 759-800 (English, with English and French summaries). MR 2680816 (2011f:46085)
  • [Tom57] Jun Tomiyama, On the projection of norm one in $ W^{\ast } $-algebras, Proc. Japan Acad. 33 (1957), 608-612. MR 0096140 (20 #2635)
  • [Web11] Samuel B. G. Webster, The path space of a higher-rank graph, Studia Math. 204 (2011), no. 2, 155-185. MR 2805537 (2012e:46119),
  • [Yan10] Dilian Yang, Endomorphisms and modular theory of 2-graph $ C^*$-algebras, Indiana Univ. Math. J. 59 (2010), no. 2, 495-520. MR 2648076 (2011m:46098),
  • [Yan12] Dilian Yang, Type III von Neumann algebras associated with 2-graphs, Bull. Lond. Math. Soc. 44 (2012), no. 4, 675-686. MR 2967236,
  • [Yan14] D. Yang, Periodic higher rank graphs revisited, to appear in J. Aust. Math. Soc..

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46L05

Retrieve articles in all journals with MSC (2010): 46L05

Additional Information

Dilian Yang
Affiliation: Department of Mathematics $&$ Statistics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario N9B 3P4, Canada

Keywords: $k$-graph algebra, cycline algebra, Cartan algebra, MASA, conditional expectation
Received by editor(s): July 22, 2015
Received by editor(s) in revised form: August 24, 2015
Published electronically: November 4, 2015
Additional Notes: The author was partially supported by an NSERC grant.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society