Advances in Operator Theory

Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property

Hiroyuki Osaka and Tamotsu Teruya

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

Let $P \subset A$ be an inclusion of unital $C^*$-algebras and $E: A \rightarrow P$ be a faithful conditional expectation of index finite type. Suppose that $E$ has the Rokhlin property. Then $\mathrm{dr}(P) \leq \mathrm{dr}(A)$ and $dim_{nuc}(P) \leq dim_{nuc}(A)$. This can be applied to Rokhlin actions of finite groups. We also show that under the same above assumption if  $A$ is exact and  pure, that is, the Cuntz semigroups $W(A)$ has strict comparison and is almost divisible, then $P$ and the basic contruction $C^*\langle A, e_P \rangle$ are also pure.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 1 (2018), 123-136.

Dates
Received: 28 March 2017
Accepted: 27 April 2017
First available in Project Euclid: 5 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512497956

Digital Object Identifier
doi:10.22034/aot.1703-1145

Mathematical Reviews number (MathSciNet)
MR3730343

Zentralblatt MATH identifier
06804320

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 46L35: Classifications of $C^*$-algebras

Keywords
Rokhlin property $C^*$-index nuclear dimension

Citation

Osaka, Hiroyuki; Teruya, Tamotsu. Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property. Adv. Oper. Theory 3 (2018), no. 1, 123--136. doi:10.22034/aot.1703-1145. https://projecteuclid.org/euclid.aot/1512497956


Export citation

References

  • B. Blackadar, Symmetries of the CAR algebras, Ann. Math. 131( 1990), 589–623.
  • B. Blackadar and D. Handelman, Dimension functions and traces on $C^*$-algebras, J. Funct. Anal. 45 (1982), 297–340.
  • S. Barlak and G. Szabo, Sequentially split $*$-homomorphoisms between $C^*$-algebras, Internat. J. Math. 27 (2016), no. 13, 1650105, 48 pp.
  • G. A. Elliott, G. Gong, H. Lin, and Z. Niu, On the classification of simple amenable $C^*$-algebras with finite decomposition rank,II, arXiv:1507.03437.
  • G. Gong, H. Lin, and Z. Niu, Classification of finite simple amenable $\mathcal{Z}$-stable $C^*$-algebras, arXiv:1501.00135.
  • U. Haagerup, Quasi-traces on exact $C^*$-algebras are traces, preprint, 1991.
  • M. Izumi, Finite group actions on $C^*$-algebras with the Rohlin property–I, Duke Math. J. 122 (2004), 233–280.
  • X. Jiang and H. Su, On a simple unital projectionless $C^*$-algebra, Amer. J. Math. 121 (1999), 359–413.
  • E. Kirchberg, On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), no. 1, 35–63.
  • E. Kirchberg and M. Rørdam, Non-simply purely infinite $C^*$-algebras, Amer. J. Math. 122 (2000), 637–666.
  • E. Kirchberg and W. Winter, Covering dimension and quasidiagonality, Internat. J. Math. 15 (2004), 63–85.
  • K. Kodaka, H. Osaka, and T. Teruya, The Rohlin property for inclusions of $C^*$-algebras with a finite Watatani index, Contemp. Math. 503 (2009), 177–195.
  • T. A. Loring, Lifting Solutions to Perturbing Problems in $C^*$-algebras, Fields Institute Monographs no. 8, American Mathematical Society, Providence RI, 1997.
  • N. Nawata, Finite group actions on certain stably projectionless $C^*$ -algebras with the Rohlin property, Trans. Amer. Math. Soc. 368 (2016), no. 1, 471–493.
  • H. Osaka and N. C. Phillips, Crossed products by finite group actions with the Rokhlin property, Math. Z. 270 (2012), 19–42.
  • H. Osaka and T. Teruya, Strongly self-absorbing property for inclusions of $C^*$-algebras with a finite Watatani index, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1685–1702.
  • V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, 146. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986.
  • N. C. Phillips, The tracial Rokhlin property for actions of finite groups on $C^*$-algebras, Amer. J. Math. 133 (2011), no. 3, 581–636 (arXiv:math.OA/0609782).
  • N. C. Phillips, Finite cyclic actions with the tracial Rokhlin property, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5271–5300. (arXiv:mathOA/0609785).
  • M. Rørdam, On the structure of simple $C^*$-algebras tensored with a UHF algebra II, J. Funct. Anal. 107 (1992), 255–269.
  • M. Rørdam, Classification of nuclear, simple $C^*$-algebras, Encycl. Math. Sci. 126, Springer, Berlin, 2002.
  • M. Rørdam, The stable and the real rank of $\mathcal{Z}$-absorbing $C^*$-algebras, Int. J. Math. 10 (2004), 1065–1084.
  • M. Rørdam and W. Winter, The Jiang-Su algebra revised, J. Reine Angew. Math. 642 (2010), 129–155.
  • L. Santiago, Crossed products by actions of finite groups with the Rokhlin property, Int. J. Math. 26 (2015), no. 7, 1550042, 31 pp.
  • T. Teruya, A characterization of normal extensions for subfactors, Proc. Amer. Math. Soc. 120 (1994), 781–783.
  • A. Tikusis, S. White, and W. Winter, Quasidiagonality of nuclear $C^*$-algebras, arXiv:1509.08318.
  • S. Wassermann, Slice map problem for $C^*$-algebras, Proc. London Math. Soc. (3), 32 (1976), no. 3, 537–559.
  • Y. Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117 pp.
  • W. Winter, Covering dimension for nuclear $C^*$-algebras I, J. Funct. Anal. 199 (2003), 535–556.
  • W. Winter, Nuclear dimension and $\mathcal{Z}$-stability of pure $C^*$-algebras, Invent. Math. 187 (2012), 259–342 (arXiv:mathOA/1006.2731v2.).
  • W. Winter and J. Zacharias, The nuclear dimension of $C^*$-algebras, Adv. Math. 224 (2010), 461–498.