Regularity of Villadsen algebras and characters on their central sequence algebras

  • Martin S. Christensen

Abstract

We show that if $A$ is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of $A$ does not admit characters.

Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast $-algebra with a unique tracial state which fails to have this property.

References

Ando, H. and Kirchberg, E., Non-commutativity of the central sequence algebra for separable non-type I $C^*$-algebras, J. Lond. Math. Soc. (2) 94 (2016), no. 1, 280–294. https://doi.org/10.1112/jlms/jdw035

>

Ara, P., Perera, F., and Toms, A. S., $K$-theory for operator algebras. Classification of $C^*$-algebras, in “Aspects of operator algebras and applications”, Contemp. Math., vol. 534, Amer. Math. Soc., Providence, RI, 2011, pp. 1--71. https://doi.org/10.1090/conm/534/10521

>

Blackadar, B., Dădărlat, M., and Rørdam, M., The real rank of inductive limit $C^*$-algebras, Math. Scand. 69 (1991), no. 2, 211–216. https://doi.org/10.7146/math.scand.a-12379

>

Blackadar, B., Robert, L., Tikuisis, A. P., Toms, A. S., and Winter, W., An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657–3674. https://doi.org/10.1090/S0002-9947-2012-05538-3

>

Coward, K. T., Elliott, G. A., and Ivanescu, C., The Cuntz semigroup as an invariant for $C^*$-algebras, J. Reine Angew. Math. 623 (2008), 161–193. https://doi.org/10.1515/CRELLE.2008.075

>

Elliott, G. A. and Niu, Z., On the classification of simple amenable $C^*$-algebras with finite decomposition rank, in “Operator algebras and their applications”, Contemp. Math., vol. 671, Amer. Math. Soc., Providence, RI, 2016, pp. 117--125. https://doi.org/10.1090/conm/671/13506

>

Elliott, G. A. and Niu, Z., The C$^*$-algebra of a minimal homeomorphism of zero mean dimension, Duke Math. J. 166 (2017), no. 18, 3569–3594. https://doi.org/10.1215/00127094-2017-0033

>

Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras I: stability, Bull. Lond. Math. Soc. 45 (2013), no. 4, 825–838. https://doi.org/10.1112/blms/bdt014

>

Ge, L. and Hadwin, D., Ultraproducts of $C^*$-algebras, in “Recent advances in operator theory and related topics (Szeged, 1999)'', Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 305--326.

Giol, J. and Kerr, D., Subshifts and perforation, J. Reine Angew. Math. 639 (2010), 107–119. https://doi.org/10.1515/CRELLE.2010.012

>

Gong, G., Jiang, X., and Su, H., Obstructions to $mathcal Z$-stability for unital simple $C^*$-algebras, Canad. Math. Bull. 43 (2000), no. 4, 418–426. https://doi.org/10.4153/CMB-2000-050-1

>

Gong, G., Lin, H., and Niu, Z., Classification of finite simple amenable $mathcal Z$-stable $C^*$-algebras, preprint arxiv:1501.00135 [math.OA], 2015.

Husemoller, D., Fibre bundles, third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4757-2261-1

>

Kirchberg, E., Central sequences in $C^*$-algebras and strongly purely infinite algebras, in “Operator Algebras: The Abel Symposium 2004”, Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 175--231. https://doi.org/10.1007/978-3-540-34197-0_10

>

Kirchberg, E. and Rørdam, M., When central sequence $C^*$-algebras have characters, Internat. J. Math. 26 (2015), no. 7, 1550049, 32. https://doi.org/10.1142/S0129167X15500494

>

Kucerovsky, D. and Ng, P. W., The corona factorization property and approximate unitary equivalence, Houston J. Math. 32 (2006), no. 2, 531–550.

Kucerovsky, D. and Ng, P. W., $S$-regularity and the corona factorization property, Math. Scand. 99 (2006), no. 2, 204–216. https://doi.org/10.7146/math.scand.a-15009

>

Kucerovsky, D. and Ng, P. W., A simple $C^ast $-algebra with perforation and the corona factorization property, J. Operator Theory 61 (2009), no. 2, 227–238.

Matui, H. and Sato, Y., Strict comparison and $mathcal Z$-absorption of nuclear $C^*$-algebras, Acta Math. 209 (2012), no. 1, 179–196. https://doi.org/10.1007/s11511-012-0084-4

>

Matui, H. and Sato, Y., Decomposition rank of UHF-absorbing $C^*$-algebras, Duke Math. J. 163 (2014), no. 14, 2687–2708. https://doi.org/10.1215/00127094-2826908

>

McDuff, D., Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443–461. https://doi.org/10.1112/plms/s3-21.3.443

>

Milnor, J. W. and Stasheff, J. D., Characteristic classes, Annals of Mathematics Studies, no. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.

Niu, Z., Mean dimension and AH-algebras with diagonal maps, J. Funct. Anal. 266 (2014), no. 8, 4938–4994. https://doi.org/10.1016/j.jfa.2014.02.010

>

Ortega, E., Perera, F., and Rørdam, M., The corona factorization property, stability, and the Cuntz semigroup of a $C^ast $-algebra, Int. Math. Res. Not. IMRN (2012), no. 1, 34–66. https://doi.org/10.1093/imrn/rnr013

>

Robert, L. and Rørdam, M., Divisibility properties for $C^*$-algebras, Proc. Lond. Math. Soc. (3) 106 (2013), no. 6, 1330–1370. https://doi.org/10.1112/plms/pds082

>

Rørdam, M., Stability of $C^*$-algebras is not a stable property, Doc. Math. 2 (1997), 375–386.

Rørdam, M., A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. https://doi.org/10.1007/BF02392697

>

Sato, Y., Discrete amenable group actions on von neumann algebras and invariant nuclear $C^ast $-subalgebras, preprint arXiv:1104.4339 [math.OA], 2011.

Tikuisis, A., White, S., and Winter, W., Quasidiagonality of nuclear $C^ast $-algebras, Ann. of Math. (2) 185 (2017), no. 1, 229–284. https://doi.org/10.4007/annals.2017.185.1.4

>

Toms, A. S., On the independence of $K$-theory and stable rank for simple $C^*$-algebras, J. Reine Angew. Math. 578 (2005), 185–199. https://doi.org/10.1515/crll.2005.2005.578.185

>

Toms, A. S., Flat dimension growth for $C^*$-algebras, J. Funct. Anal. 238 (2006), no. 2, 678–708. https://doi.org/10.1016/j.jfa.2006.01.010

>

Toms, A. S., On the classification problem for nuclear $C^ast $-algebras, Ann. of Math. (2) 167 (2008), no. 3, 1029–1044. https://doi.org/10.4007/annals.2008.167.1029

>

Toms, A. S., Comparison theory and smooth minimal $C^*$-dynamics, Comm. Math. Phys. 289 (2009), no. 2, 401–433. https://doi.org/10.1007/s00220-008-0665-4

>

Toms, A. S. and Winter, W., The Elliott conjecture for Villadsen algebras of the first type, J. Funct. Anal. 256 (2009), no. 5, 1311–1340. https://doi.org/10.1016/j.jfa.2008.12.015

>

Villadsen, J., Simple $C^*$-algebras with perforation, J. Funct. Anal. 154 (1998), no. 1, 110–116. https://doi.org/10.1006/jfan.1997.3168

>

Villadsen, J., On the stable rank of simple $C^ast $-algebras, J. Amer. Math. Soc. 12 (1999), no. 4, 1091–1102. https://doi.org/10.1090/S0894-0347-99-00314-8

>

Published
2018-08-01
How to Cite
Christensen, M. (2018). Regularity of Villadsen algebras and characters on their central sequence algebras. MATHEMATICA SCANDINAVICA, 123(1), 121-141. https://doi.org/10.7146/math.scand.a-104840
Section
Articles