×

The nuclear dimension of \(C^{*}\)-algebras. (English) Zbl 1201.46056

The authors introduce nuclear dimension for \(C^*\)-algebras as follows. A \(C^*\)-algebra \(A\) has nuclear dimension at most \(n\) if there exists a net \((F_\lambda,\psi_\lambda,\varphi_\lambda)_{\lambda\in\Lambda}\), where \(F_\lambda\) are finite-dimensional \(C^*\)-algebras, \(\psi_\lambda:A\to F_\lambda\) and \(\varphi_\lambda:F_\lambda\to A\) are completely positive maps such that 8mm
(i)
\(\varphi_\lambda\circ\psi_\lambda(a)\to a\) uniformly on finite subsets of \(A\);
(ii)
\(\|\psi_\lambda\|\leq 1\);
(iii)
for each \(\lambda\), there is a decomposition \(F_\lambda=F_\lambda^{(1)}\oplus\cdots\oplus F_\lambda^{(n)}\) such that each \(\varphi_\lambda|_{F_\lambda^{(i)}}\), \(i=1,\dots, n\), is a contraction of order zero (the latter means that it respects orthogonality: \(\varphi\) is of order zero if, for all positive \(a, b\), \(a\perp b\) implies \(\varphi(a)\perp\varphi(b)\)).
It is shown that nuclear dimension behaves well with respect to inductive limits, tensor products, hereditary subalgebras, quotients and extensions. The evaluation of the nuclear dimension for many \(C^*\)-algebras is given. In particular, it is shown that the nuclear dimension is finite for all UCT Kirchberg algebras, and that, for a discrete metric space of bounded geometry, the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.

MSC:

46L35 Classifications of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
46L85 Noncommutative topology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bell, G.; Dranishnikov, A., Asymptotic dimension, Topology Appl., 155, 1265-1296 (2008) · Zbl 1149.54017
[2] Blanchard, E.; Kirchberg, E., Nonsimple purely infinite \(C^\ast \)-algebras: the Hausdorff case, J. Funct. Anal., 207, 461-513 (2004) · Zbl 1048.46049
[3] Brown, L. G., Stable isomorphism of hereditary subalgebras of \(C^\ast \)-algebras, Pacific J. Math., 71, 335-348 (1977) · Zbl 0362.46042
[4] Brown, L. G.; Pedersen, G. K., \(C^\ast \)-algebras of real rank zero, J. Funct. Anal., 99, 131-149 (1991) · Zbl 0776.46026
[5] Connes, A., Compact metric spaces, Fredholm modules and hyperfiniteness, Ergodic Theory Dynam. Systems, 9, 207-220 (1989) · Zbl 0718.46051
[6] Connes, A., Noncommutative Geometry (1994), Academic Press Inc.: Academic Press Inc. San Diego, CA · Zbl 0681.55004
[7] Elliott, G. A.; Gong, G.; Li, L., On the classification of simple inductive limit \(C^\ast \)-algebras II: the isomorphism theorem, Invent. Math., 168, 249-320 (2007) · Zbl 1129.46051
[8] Evans, D. E., On \(O_n\), Publ. Res. Inst. Math. Sci., 16, 915-927 (1980) · Zbl 0461.46042
[10] Hurewicz, W.; Wallman, H., Dimension Theory (1951), Princeton University Press: Princeton University Press Princeton · JFM 67.1092.03
[11] Jiang, X.; Su, H., On a simple unital projectionless \(C^\ast \)-algebra, Amer. J. Math., 121, 359-413 (1999) · Zbl 0923.46069
[12] Kirchberg, E., Exact \(C^\ast \)-algebras, tensor products, and the classification of purely infinite \(C^\ast \)-algebras, (Proceedings of the International Congress of Mathematicians. Proceedings of the International Congress of Mathematicians, Zürich, 1994 (1994), Birkhäuser: Birkhäuser Basel), 943-954 · Zbl 0897.46057
[13] Kirchberg, E., On the existence of traces on exact stably projectionless simple \(C^\ast \)-algebras, (Fields Inst. Commun., vol. 13 (1995)), 171-172 · Zbl 0894.46043
[14] Kirchberg, E., Central sequences in \(C^\ast \)-algebras and strongly purely infinite \(C^\ast \)-algebras, Abel Symp., 1, 175-231 (2006) · Zbl 1118.46054
[15] Kirchberg, E.; Rørdam, M., Non-simple purely infinite \(C^\ast \)-algebras, Amer. J. Math., 122, 637-666 (2000) · Zbl 0968.46042
[16] Kirchberg, E.; Winter, W., Covering dimension and quasidiagonality, Internat. J. Math., 15, 63-85 (2004) · Zbl 1065.46053
[17] Kribs, D., Inductive limit algebras from periodic weighted shifts on Fock space, New York J. Math., 8, 145-159 (2002) · Zbl 1026.46041
[18] Lin, Q.; Phillips, N. C., Direct limit decomposition for \(C^\ast \)-algebras of minimal diffeomorphisms, (Operator Algebras and Applications. Operator Algebras and Applications, Adv. Stud. Pure Math., vol. 38 (2004), Math. Soc. Japan: Math. Soc. Japan Tokyo), 107-133 · Zbl 1074.46047
[19] Loring, T., Lifting Solutions to Perturbing Problems in \(C^\ast \)-algebras, Fields Inst. Monogr., vol. 8 (1997), AMS: AMS Providence, RI · Zbl 1155.46310
[21] Rieffel, M., Dimension and stable rank in the \(K\)-theory of \(C^\ast \)-algebras, Proc. London Math. Soc., 46, 301-333 (1983) · Zbl 0533.46046
[22] Roe, J., Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31 (2003), AMS · Zbl 1042.53027
[23] Rørdam, M., On the structure of simple \(C^\ast \)-algebras tensored with a UHF algebra, II, J. Funct. Anal., 107, 255-269 (1992) · Zbl 0810.46067
[24] Rørdam, M., Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math., 440, 175-200 (1993) · Zbl 0783.46031
[25] Rørdam, M., Classification of Nuclear \(C^\ast \)-Algebras, Encyclopaedia Math. Sci., vol. 126 (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 1016.46037
[26] Rørdam, M., A simple \(C^\ast \)-algebra with a finite and an infinite projection, Acta Math., 191, 109-142 (2003) · Zbl 1072.46036
[27] Rørdam, M., The stable and the real rank of \(Z\)-absorbing \(C^\ast \)-algebras, Int. J. Math., 15, 1065-1084 (2004) · Zbl 1077.46054
[28] Rørdam, M.; Winter, W., The Jiang-Su algebra revisited (2008), J. Reine Angew. Math., in press · Zbl 1209.46031
[29] Skalski, A.; Zacharias, J., On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules (2006), Rocky Mountain J. Math., in press
[31] Toms, A. S., On the classification problem for nuclear \(C^\ast \)-algebras, Ann. of Math. (2), 167, 1059-1074 (2008)
[32] Toms, A. S.; Winter, W., Minimal dynamics and \(K\)-theoretic rigidity: Elliott’s conjecture (2009), preprint · Zbl 1280.46046
[33] Toms, A. S.; Winter, W., Minimal dynamics and the classification of \(C^\ast \)-algebras, Proc. Natl. Acad. Sci., 106, 40, 16942-16943 (2009) · Zbl 1203.46046
[34] Winter, W., Covering dimension for nuclear \(C^\ast \)-algebras, J. Funct. Anal., 199, 535-556 (2003) · Zbl 1026.46049
[35] Winter, W., Decomposition rank of subhomogeneous \(C^\ast \)-algebras, Proc. London Math. Soc., 89, 427-456 (2004) · Zbl 1081.46049
[36] Winter, W., On topologically finite dimensional simple \(C^\ast \)-algebras, Math. Ann., 332, 843-878 (2005) · Zbl 1089.46039
[37] Winter, W., Simple \(C^\ast \)-algebras with locally finite decomposition rank, J. Funct. Anal., 243, 394-425 (2007) · Zbl 1121.46047
[38] Winter, W., Localizing the Elliott conjecture at strongly self-absorbing \(C^\ast \)-algebras (2007), preprint
[39] Winter, W., Decomposition rank and \(Z\)-stability, Invent. Math., 179, 229-301 (2010) · Zbl 1194.46104
[40] Winter, W., Covering dimension for nuclear \(C^\ast \)-algebras II, Trans. Amer. Math. Soc., 361, 4143-4167 (2009) · Zbl 1178.46070
[41] Winter, W.; Zacharias, J., Completely positive maps of order zero, Münster J. Math., 2, 311-324 (2009) · Zbl 1190.46042
[42] Wolff, M., Disjointness preserving operators on \(C^\ast \)-algebras, Arch. Math., 62, 248-253 (1994) · Zbl 0803.46069
[43] Yu, G., The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2), 147, 325-355 (1998) · Zbl 0911.19001
[44] Zacharias, J., On the invariant translation approximation property for discrete groups, Proc. Amer. Math. Soc., 134, 1909-1916 (2006) · Zbl 1099.46038
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.