Decomposable approximations and approximately finite dimensional \(C^*\)-algebras. (English) Zbl 1442.46046

A \(C^*\)-algebra \(A\) is said to be nuclear if, for every other \(C^*\)-algebra \(B\), there is a unique \(C^*\)-norm on the algebraic tensor product \(A\odot B\). This important property is equivalent to the completely positive approximation property (CPAP): the identity on \(A\) can be approximately factored as the composition of completely positive, contractive (cpc) maps \(A\to F\) and \(F\to A\) with \(F\) a finite-dimensional \(C^*\)-algebra.
One obtains interesting subclasses of nuclear \(C^*\)-algebras by considering strengthened versions of the CPAP where one requires additional properties of the map \(F\to A\). For example, one says that \(A\) has decomposition rank at most \(n\) if the identity on \(A\) can be approximately factored through cpc-maps \(A\to F\) and \(\varphi\colon F\to A\) such that \(F\) decomposes as a direct sum of \(n+1\) summands and the restriction of \(\varphi\) to each summand is orthogonality-preserving. The decomposition rank is a noncommutative generalization of covering dimension introduced in [E. Kirchberg and W. Winter, Int. J. Math. 15, No. 1, 63–85 (2004; Zbl 1065.46053)], and it plays an important role in the current structure and classification theory of simple, nuclear \(C^*\)-algebras.
In [I. Hirshberg et al., Adv. Math. 230, No. 3, 1029–1039 (2012; Zbl 1256.46019)] it was shown that nuclear \(C^*\)-algebras automatically enjoy a strengthened versions of the CPAP: one may additionally assume that the cpc-map \(F\to A\) is a convex combination of orthogonality-preserving cpc-maps, but the number of summands may grow with increasing precision of the approximation.
It is thus natural to consider the combination of these strengthenings of the CPAP: Given \(n\), when does a \(C^*\)-algebra \(A\) admit approximate factorizations of the identity through cpc-maps \(A\to F\) and \(\varphi\colon F\to A\) such that \(\varphi\) is a convex combination of at most \(n\) orthogonality-preserving cpc-maps? The present paper shows that this happens if and only if \(A\) has decomposition rank zero, which for separable \(C^*\)-algebras means precisely that \(A\) is an AF-algebra.


46L35 Classifications of \(C^*\)-algebras
46B28 Spaces of operators; tensor products; approximation properties
Full Text: DOI arXiv


[1] Blackadar, B.Operator Algebras Encyclopaedia of Mathematical Sciences vol. 122 (Springer-Verlag, Berlin, 2006). Theory of C*-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III. doi:10.1007/3-540-28517-2 · Zbl 1092.46003
[2] Bratteli, O., Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc., 171, 195-234, (1972) · Zbl 0264.46057
[3] Busby, R., Double centralizers and extensions of C*-algebras, Trans. Amer. Math. Soc., 132, 79-99, (1968) · Zbl 0165.15501
[4] Choi, M. D. and Effros, E.Nuclear C*-algebras and the approximation property. Amer. J. Math.100(1) (1978), 61-79. doi:10.2307/2373876 · Zbl 0397.46054
[5] Elliott, G. and Toms, A.Regularity properties in the classification program for separable amenable C*-algebras. Bull. Amer. Math. Soc.45(2) (2008), 229-245. doi:10.1090/S0273-0979-08-01199-3 · Zbl 1151.46048
[6] Farah, I. and Katsura, T.Nonseparable UHF algebras I: Dixmier’s problem. Adv. Math.225(3) (2010), 1399-1430. doi:10.1016/j.aim.2010.04.006 · Zbl 1202.46063
[7] Hirshberg, I., Kirchberg, E. and White, S.Decomposable approximations of nuclear C*-algebras. Adv. Math.230(3) (2012), 1029-1039. doi:10.1016/j.aim.2012.03.028 · Zbl 1256.46019
[8] Kirchberg, E., C*-nuclearity implies CPAP, Math. Nachr., 76, 203-212, (1977) · Zbl 0383.46011
[9] Kirchberg, E. and Winter, W.Covering dimension and quasidiagonality. Internat. J. Math.15(1) (2004), 63-85. doi:10.1142/S0129167X04002119 · Zbl 1065.46053
[10] Matui, H. and Sato, Y.Strict comparison and \(\mathcal{Z}\)-absorption of nuclear C*-algebras. Acta Math.209(1) (2012), 179-196. doi:10.1007/s11511-012-0084-4 · Zbl 1277.46028
[11] Matui, H. and Sato, Y.Decomposition rank of UHF-absorbing C*-algebras. Duke Math. J.163(14) (2014), 2687-2708. doi:10.1215/00127094-2826908 · Zbl 1317.46041
[12] Robert, L., Nuclear dimension and n-comparison, Münster J. Math., 4, 65-71, (2011) · Zbl 1248.46040
[13] Sato, Y., White, S. and Winter, W.Nuclear dimension and \(\mathcal{Z}\)-stability. Invent. Math.202(2) (2015), 893-921. doi:10.1007/s00222-015-0580-1 · Zbl 1350.46040
[14] Tikuisis, A. and Winter, W.Decomposition rank of \(\mathcal{Z}\)-stable C*-algebras. Anal. PDE.7(3) (2014), 673-700. doi:10.2140/apde.2014.7.673 · Zbl 1303.46048
[15] Winter, W., Covering dimension for nuclear C*-algebras, J. Funct. Anal., 199, 535-556, (2003) · Zbl 1026.46049
[16] Winter, W., Decomposition rank and \(\mathcal{Z}\)-stability, Invent. Math., 179, 229-301, (2010) · Zbl 1194.46104
[17] Winter, W., Nuclear dimension and \(\mathcal{Z}\)-stability of pure C*-algebras, Invent. Math., 187, 259-342, (2012) · Zbl 1280.46041
[18] Winter, W. and Zacharias, J.Completely positive maps of order zero. Münster J. Math.2 (2009), 311-324. · Zbl 1190.46042
[19] Winter, W. and Zacharias, J.The nuclear dimension of C*-algebras. Adv. Math.224(2) (2010), 461-498. doi:10.1016/j.aim.2009.12.005 · Zbl 1201.46056
[20] Wolff, M., Disjointness preserving operators on C*-algebras, Arch. Math. (Basel), 62, 248-253, (1994) · Zbl 0803.46069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.