Rocky Mountain Journal of Mathematics

Continuous-trace $k$-graph $C^*$-algebras

Danny Crytser

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the Drinen-Tomforde desingularization. A characterization of continuous-trace AF $C^*$-algebras is obtained. Partial results are given to characterize higher-rank graphs that yield $C^*$-algebras with continuous trace.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1511-1535.

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936034

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1511

Mathematical Reviews number (MathSciNet)
MR3866557

Zentralblatt MATH identifier
06958790

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Keywords
$C^*$-algebras graphs groupoids

Citation

Crytser, Danny. Continuous-trace $k$-graph $C^*$-algebras. Rocky Mountain J. Math. 48 (2018), no. 5, 1511--1535. doi:10.1216/RMJ-2018-48-5-1511. https://projecteuclid.org/euclid.rmjm/1539936034


Export citation

References

  • J. Brown and D. Yang, Cartan subalgebras of topological graph algebras and $k$-graph $C^*$-algebras, arXiv:1511.01517.
  • L.O. Clark, Classifying the types of principal groupoid $C^*$-algebras, J. Oper. Th. 57 (2007), 251–266.
  • D. Drinen, Viewing AF algebras as graph algebras, Proc. Amer. Math. Soc. 128 (1999), 1991–2000.
  • D. Drinen and M. Tomforde, The ${C^*}$-algebras of arbitrary graphs, Rocky Mountain J. Math. 35 (2005), 105–135.
  • D.G. Evans and A. Sims, When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?, J. Funct. Anal. 263 (2012), 183–215.
  • C. Farthing, Removing sources from higher-rank graphs, J. Oper. Th. 60 (2008), 165–198.
  • G. Goehle, Groupoid crossed products, Ph.D. dissertation, Dartmouth College, Hanover, 2009.
  • ––––, Groupoid $C^*$-algebras with Hausdorff spectrum, Bull. Australian Math. Soc. 88 (2013), 232–242.
  • R. Hazlewood, Categorising the operator algebras of groupoids and higher-rank graphs, Ph.D. dissertation, The University of New South Wales, Kensington, 2013.
  • A. Kumjian and D. Pask, Higher-rank graph $C^*$-algebras, New York J. Math. 6 (2001), 1–20.
  • A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math 184 (1998), 161–174.
  • A. Kumjian, D. Pask, I. Raeburn and J. Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505–541.
  • S. Lalonde and D. Milan, Amenability and uniqueness for groupoids associated with inverse semigroups, Semigroup Forum (2016), 1–24.
  • P. Muhly, J. Renault and D. Williams, Continuous-trace groupoid ${C^*}$-algebras, III, Trans. Amer. Math. Soc. 348 (1996), 3621–3641.
  • G.K. Pedersen, ${C^*}$-algebras and their automorphism groups, Academic Press, San Diego, 1979.
  • I. Raeburn, Graph algebras, American Mathematical Society, Providence, 2005.
  • I. Raeburn and D.P. Williams, Morita equivalence and continuous-trace ${C^*}$-algebras, Math. Surv. Mono. 60 (1998).
  • J. Renault, A groupoid approach to $C^*$-algebras, Lect. Notes Math. 793 (1980).
  • J. Tyler, Every AF-algebra is Morita equivalent to a graph algebra, Bull. Australian Math. Soc. 69 (2004), 237–240.
  • S. Webster, The path space of a higher-rank graph, Stud. Math. 204 (2011), 155–185.