## Duke Mathematical Journal

### Decomposition rank of UHF-absorbing $\mathrm{C}^{*}$-algebras

#### Abstract

Let $A$ be a unital separable simple $\mathrm{C}^{*}$-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

https://projecteuclid.org/euclid.dmj/1414762068

Digital Object Identifier
doi:10.1215/00127094-2826908

Mathematical Reviews number (MathSciNet)
MR3273581

Zentralblatt MATH identifier
1317.46041

#### 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

#### 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.