Duke Mathematical Journal

Decomposition rank of UHF-absorbing C-algebras

Hiroki Matui and Yasuhiko Sato

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 A be a unital separable simple C-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one. We then prove that A is nuclear, quasidiagonal, and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF algebra has tracial rank zero. Using this result, we obtain a counterexample to the Powers–Sakai conjecture.

Article information

Source
Duke Math. J., Volume 163, Number 14 (2014), 2687-2708.

Dates
First available in Project Euclid: 31 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1414762068

Digital Object Identifier
doi:10.1215/00127094-2826908

Mathematical Reviews number (MathSciNet)
MR3273581

Zentralblatt MATH identifier
1317.46041

Subjects
Primary: 46L06: Tensor products of $C^*$-algebras 46L35: Classifications of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Keywords
$C*$-algebras Jiang–Su algebra decomposition rank

Citation

Matui, Hiroki; Sato, Yasuhiko. Decomposition rank of UHF-absorbing $\mathrm{C}^{*}$ -algebras. Duke Math. J. 163 (2014), no. 14, 2687--2708. doi:10.1215/00127094-2826908. https://projecteuclid.org/euclid.dmj/1414762068


Export citation

References

  • [1] C. A. Akemann, J. Anderson, and G. K. Pedersen, Excising states of $\mathrm{C}^{*}$-algebras, Canad. J. Math. 38 (1986), 1239–1260.
  • [2] B. Blackadar, “Comparison theory for simple $\mathrm{C}^{*}$-algebras” in Operator Algebras and Applications, Vol. I, London Math. Soc. Lecture Note Ser. 135, Cambridge Univ. Press, Cambridge, 1988, 21–54.
  • [3] O. Bratteli, G. A. Elliott, and R. H. Herman, On the possible temperatures of a dynamical system, Comm. Math. Phys. 74 (1980), 281–295.
  • [4] N. P. Brown, AF embeddability of crossed products of AF algebras by the integers, J. Funct. Anal. 160 (1998), 150–175.
  • [5] N. P. Brown and N. Ozawa, $\mathrm{C}^{*}$-algebras and Finite-dimensional Approximations, Grad. Stud. Math. 88, Amer. Math. Soc., Providence, 2008.
  • [6] A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. Éc. Norm. Supér. (4) 8 (1975), 383–419.
  • [7] A. Connes, Classification of injective factors: Cases $\mathit{II}_{1}$, $\mathit{II}_{\infty}$, $\mathit{III}_{\lambda}$, $\lambda\neq1$, Ann. of Math. (2) 104 (1976), 73–115.
  • [8] A. Connes, Periodic automorphisms of the hyperfinite factor of type II$_{1}$, Acta Sci. Math. (Szeged) 39 (1977), 39–66.
  • [9] U. Haagerup, A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space, J. Funct. Anal. 62 (1985), 160–201.
  • [10] X. Jiang and H. Su, On a simple unital projectionless $\mathrm{C}^{*}$-algebra, Amer. J. Math. 121 (1999), 359–413.
  • [11] E. Kirchberg, The classification of purely infinite $\mathrm{C}^{*}$-algebras using Kasparov’s theory, preprint, 1994.
  • [12] E. Kirchberg and M. Rørdam, Infinite non-simple $\mathrm{C}^{*}$-algebras: Absorbing the Cuntz algebras $\mathcal{O}_{\infty}$, Adv. Math. 167 (2002), 195–264.
  • [13] E. Kirchberg and M. Rørdam, Central sequence $\mathrm{C}^{*}$-algebras and tensorial absorption of the Jiang-Su algebra, to appear in J. Reine Angew. Math., preprint, arXiv:1209.5311v5 [math.OA].
  • [14] E. Kirchberg and W. Winter, Covering dimension and quasidiagonality, Internat. J. Math. 15 (2004), 63–85.
  • [15] A. Kishimoto, Non-commutative shifts and crossed products, J. Funct. Anal. 200 (2003), 281–300.
  • [16] H. Lin, The tracial topological rank of $\mathrm{C}^{*}$-algebras, Proc. Lond. Math. Soc. (3) 83 (2001), 199–234.
  • [17] H. Lin, Tracially AF $\mathrm{C}^{*}$-algebras, Trans. Amer. Math. Soc. 353 (2001), no. 2, 693–722.
  • [18] H. Lin, AF-embeddings of the crossed products of AH-algebras by finitely generated abelian groups Int. Math. Res. Pap. IMRP 2008, no. 3, art. ID rpn007.
  • [19] H. Lin and Z. Niu, Lifting $KK$-elements, asymptotic unitary equivalence and classification of simple $\mathrm{C}^{*}$-algebras, Adv. Math. 219 (2008), 1729–1769.
  • [20] H. Lin and Z. Niu, The range of a class of classifiable separable simple amenable $\mathrm{C}^{*}$-algebras, J. Funct. Anal. 260 (2011), 1-29.
  • [21] T. A. Loring, Lifting Solutions to Perturbing Problems in $\mathrm{C}^{*}$-algebras, Fields Inst. Monogr. 8, Amer. Math. Soc., Providence, 1997.
  • [22] H. Matui and Y. Sato, Strict comparison and $\mathcal{Z} $-absorption of nuclear $\mathrm{C}^{*}$-algebras, Acta Math. 209 (2012), 179–196.
  • [23] H. Matui and Y. Sato, $\mathcal{Z} $-stability of crossed products by strongly outer actions, Comm. Math. Phys. 314 (2012), 193–228.
  • [24] H. Matui and Y. Sato, $\mathcal{Z}$-stability of crossed products by strongly outer actions, II, to appear in Amer. J. Math., preprint, arXiv:1205.1590v2 [math.OA].
  • [25] S. Popa, A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras, J. Operator Theory 16 (1986), 261–272.
  • [26] S. Popa, On local finite-dimensional approximation of $\mathrm{C}^{*}$-algebras, Pacific J. Math. 181 (1997), 141–158.
  • [27] R. T. Powers and S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1974/75), 273–288.
  • [28] M. Rørdam, On the structure of simple $\mathrm{C}^{*}$-algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991), 1–17.
  • [29] M. Rørdam, The stable and the real rank of $\mathcal{Z}$-absorbing $\mathrm{C}^{*}$-algebras, Internat. J. Math. 15 (2004), 1065–1084.
  • [30] Y. Sato, The Rohlin property for automorphisms of the Jiang-Su algebra, J. Funct. Anal. 259 (2010), 453–476.
  • [31] Y. Sato, Trace spaces of simple nuclear $\mathrm{C}^{*}$-algebras with finite-dimensional extreme boundary, preprint, arXiv:1209.3000v1 [math.OA].
  • [32] G. Szabó, The Rokhlin dimension of topological $\mathbb{Z} ^{m}$-actions, preprint, arXiv:1308.5418v4 [math.OA].
  • [33] A. S. Toms, S. White, and W. Winter, $\mathcal{Z}$-stability and finite dimensional tracial boundaries, preprint, arXiv:1209.3292v2 [math.OA].
  • [34] W. Winter, Simple $\mathrm{C}^{*}$-algebras with locally finite decomposition rank, J. Funct. Anal. 243 (2007), 394–425.
  • [35] W. Winter, Decomposition rank and $\mathcal{Z} $-stability, Invent. Math. 179 (2010), 229–301.
  • [36] W. Winter, Classifying crossed product $C^{*}$-algebras, preprint, arXiv:1308.5084v1 [math.OA].
  • [37] W. Winter and J. Zacharias, The nuclear dimension of $\mathrm{C}^{*}$-algebras, Adv. Math. 224 (2010), 461–498.