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The generator rank of subhomogeneous $C^*\!$-algebras

Published online by Cambridge University Press:  14 June 2022

Hannes Thiel*
Affiliation:
Department of Mathematics, Kiel University, Heinrich-Hecht-Platz 6, 24118Kiel, Germany URL: www.hannesthiel.org

Abstract

We compute the generator rank of a subhomogeneous $C^*\!$ -algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every $\mathcal {Z}$ -stable approximately subhomogeneous algebra has generator rank one, which means that a generic element in such an algebra is a generator.

This leads to a strong solution of the generator problem for classifiable, simple, nuclear $C^*\!$ -algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear $C^*\!$ -algebras.

Type
Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The author was partially supported by the Deutsche Forschungsgemeinschaft (DFG; German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587 (Mathematics Münster: Dynamics-Geometry-Structure).

References

Alfsen, E. M., Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, 57, Springer, New York–Heidelberg, 1971.CrossRefGoogle Scholar
Beggs, E. J. and Evans, D. E., The real rank of algebras of matrix valued functions . Int. J. Math. 2(1991), 131138.CrossRefGoogle Scholar
Blackadar, B., Operator algebras: theory of C*-algebras and von Neumann algebras, operator algebras and non-commutative geometry, III, Encyclopaedia of Mathematical Sciences, 122, Springer, Berlin, 2006.CrossRefGoogle Scholar
Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York–London, 1972.Google Scholar
Brown, L. G., On higher real and stable ranks for  $CCRC^{\ast}$ -algebras . Trans. Amer. Math. Soc. 368(2016), 74617475.CrossRefGoogle Scholar
Brown, L. G. and Pedersen, G. K., Limits and  $C^\ast\!$ -algebras of low rank or dimension . J. Operator Theory 61(2009), 381417.Google Scholar
Dadarlat, M., Continuous fields of  $C^\ast\!$ -algebras over finite dimensional spaces . Adv. Math. 222(2009), 18501881.CrossRefGoogle Scholar
Elliott, G. A., Gong, G., Lin, H., and Niu, Z., On the classification of simple amenable $C^{\ast}\!\!$ -algebras with finite decomposition rank, II. Preprint, 2016, arXiv:1507.03437 [math.OA].CrossRefGoogle Scholar
Farah, I., Hart, B., Lupini, M., Robert, L., Tikuisis, A., Vignati, A., and Winter, W., Model theory of  $C^\ast\!$ -algebras . Mem. Amer. Math. Soc. 271(2021), viii+127.Google Scholar
Farah, I. and Katsura, T., Nonseparable UHF algebras I: Dixmier’s problem . Adv. Math. 225(2010), 13991430.CrossRefGoogle Scholar
Fell, J. M. G., The structure of algebras of operator fields . Acta Math. 106(1961), 233280.CrossRefGoogle Scholar
Gong, G., Lin, H., and Niu, Z., A classification of finite simple amenable $\mathbf{\mathcal{Z}}$ -stable $C^\ast\!\!$ -algebras, II: $C^\ast\!\!$ -algebras with rational generalized tracial rank one. C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), 451539.Google Scholar
Luukkainen, J., Approximating continuous maps of metric spaces into manifolds by embeddings . Math. Scand. 49(1981), 6185.CrossRefGoogle Scholar
Meinreken, E., Group actions on manifolds, Lecture Notes, University of Toronto, Spring 2003.Google Scholar
Nagisa, M., Single generation and rank of C*-algebras . In: Operator algebras and applications, Advanced Studies in Pure Mathematics, 38, Mathematical Society of Japan, Tokyo, 2004, pp. 135143.CrossRefGoogle Scholar
Ng, P. W. and Winter, W., A note on subhomogeneous  $C^\ast\!$ -algebras . C. R. Math. Acad. Sci. Soc. R. Can. 28(2006), 9196.Google Scholar
Pears, A. R., Dimension theory of general spaces, Cambridge University Press, Cambridge–New York–Melbourne, 1975.Google Scholar
Phillips, N. C., Recursive subhomogeneous algebras . Trans. Amer. Math. Soc. 359(2007), 45954623.CrossRefGoogle Scholar
Thiel, H., The topological dimension of type I C*-algebras . In: Operator algebra and dynamics, Springer Proceedings in Mathematics and Statistics, 58, Springer, Heidelberg, 2013, pp. 305328.CrossRefGoogle Scholar
Thiel, H., Generators in $\mathbf{\mathcal{Z}}$ -stable  $C^\ast\!$ -algebras of real rank zero. J. Noncommut. Geom. Preprint, 2020, arXiv:2006.08404 [math.OA].Google Scholar
Thiel, H., The generator rank of  $C^\ast\!$ -algebras . J. Funct. Anal. 280(2021), 108874, 34 pp.CrossRefGoogle Scholar
Thiel, H. and Winter, W., The generator problem for  $\mathbf{\mathcal{Z}}$ -stable  $C^\ast\!$ -algebras . Trans. Amer. Math. Soc. 366(2014), 23272343.CrossRefGoogle Scholar
Tikuisis, A., White, S., and Winter, W., Quasidiagonality of nuclear  $C^\ast\!$ -algebras . Ann. Math. 185(2017), 229284.CrossRefGoogle Scholar
van Mill, J., The infinite-dimensional topology of function spaces, North-Holland Mathematical Library, 64, North-Holland, Amsterdam, 2001.Google Scholar
Villadsen, J., On the stable rank of simple  $C^\ast\!$ -algebras . J. Amer. Math. Soc. 12(1999), 10911102.CrossRefGoogle Scholar

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