Rocky Mountain Journal of Mathematics

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

Danny Crytser

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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

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

First available in Project Euclid: 19 October 2018

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Zentralblatt MATH identifier

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

$C^*$-algebras graphs groupoids


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.

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  • 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.