Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-11T03:32:53.789Z Has data issue: false hasContentIssue false
  • English
  • Français

RATIONAL $\boldsymbol {K}$-STABILITY OF CONTINUOUS $\boldsymbol {C(X)}$-ALGEBRAS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  10 May 2022

APURVA SETH  and
PRAHLAD VAIDYANATHAN
APURVA SETH
Affiliation:
Department of Mathematics, IISER Bhopal, Bhopal ByPass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India e-mail: [email protected]
PRAHLAD VAIDYANATHAN*
Affiliation:
Department of Mathematics, IISER Bhopal, Bhopal ByPass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the properties of being rationally K-stable passes from the fibres of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers.

Keywords

nonstable K-theoryC*-algebras

MSC classification

Primary: 46L85: Noncommutative topology
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 115 , Issue 1 , August 2023 , pp. 119 - 144
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Robert Yuncken

The first named author is supported by UGC Junior Research Fellowship No. 1229, and the second named author was partially supported by the SERB (Grant No. MTR/2020/000385).

References

Barlak, S. and Szabó, G., ‘Sequentially split $\ast$ -homomorphisms between ${C}^{\ast }$ -algebras’, Internat. J. Math. 27(13) (2016), 1650105, 48.CrossRefGoogle Scholar
Dadarlat, M., ‘Continuous fields of ${C}^{\ast }$ -algebras over finite dimensional spaces’, Adv. Math. 222(5) (2009), 1850–1881.CrossRefGoogle Scholar
Dold, A., ‘Partitions of unity in the theory of fibrations’, Ann. of Math. (2) 78 (1963), 223–255.CrossRefGoogle Scholar
Engelking, R., Dimension Theory (North-Holland, Amsterdam, 1978).Google Scholar
Farjoun, E. D. and Schochet, C. L., ‘Spaces of sections of Banach algebra bundles’, J. K-Theory 10(2) (2012), 279–298.CrossRefGoogle Scholar
Federer, H., ‘A study of function spaces by spectral sequences’, Trans. Amer. Math. Soc. 82 (1956), 340–361.CrossRefGoogle Scholar
Freudenthal, H., ‘Entwicklungen von Räumen und ihren Gruppen’, Compos. Math. 4 (1937), 145–234.Google Scholar
Gardella, E., ‘Rokhlin dimension for compact group actions’, Indiana Univ. Math. J. 66(2) (2017), 659–703.CrossRefGoogle Scholar
Handelman, D., ‘ ${K}_0$ of von Neumann and AF ${C}^{\ast }$ algebras’, Q. J. Math. 29(116) (1978), 427–441.CrossRefGoogle Scholar
Hilton, P., Mislin, G. and Roitberg, J., Localization of Nilpotent Groups and Spaces (North-Holland, Amsterdam, 1975).Google Scholar
Gardella, E., Hirshberg, I. and Santiago, L., ‘Rokhlin dimension: duality, tracial properties, and crossed products’, Ergodic Theory Dynam. Systems 41(2) (2021), 408–460.CrossRefGoogle Scholar
Hirshberg, I., Winter, W. and Zacharias, J., ‘Rokhlin dimension and ${C}^{\ast }$ -dynamics’, Comm. Math. Phys. 335(2) (2015), 637–670.CrossRefGoogle Scholar
Kasparov, G. G., ‘Equivariant $\mathrm{KK}$ -theory and the Novikov conjecture’, Invent. Math. 91(1) (1988), 147–201.CrossRefGoogle Scholar
Kirchberg, E. and Wassermann, S., ‘Operations on continuous bundles of ${C}^{\ast }$ -algebras’, Math. Ann. 303(4) (1995), 677–697.CrossRefGoogle Scholar
Mardešić, S., ‘On covering dimension and inverse limits of compact spaces’, Illinois J. Math. 4 (1960), 278–291.CrossRefGoogle Scholar
May, J. P. and Ponto, K., More Concise Algebraic Topology: Localization, Completion, and Model Categories (University of Chicago Press, Chicago, IL, 2012).Google Scholar
Palais, R. S., The Classification of $G$ -Spaces, Memoirs of the American Mathematical Society, 36 (American Mathematical Society, Providence, RI, 1960).CrossRefGoogle Scholar
Pedersen, G. K., ‘Pullback and pushout constructions in ${C}^{\ast }$ -algebra theory’, J. Funct. Anal. 167(2) (1999), 243–344.CrossRefGoogle Scholar
Phillips, N. C., Lupton, G., Schochet, C. L. and Smith, S. B., ‘Banach algebras and rational homotopy theory’, Trans. Amer. Math. Soc. 361(1) (2009), 267–295.Google Scholar
Rieffel, M. A., ‘The homotopy groups of the unitary groups of noncommutative tori’, J. Operator Theory 17(2) (1987), 237–254.Google Scholar
Schochet, C., ‘Topological methods for ${C}^{\ast }$ -algebras. III. Axiomatic homology’, Pacific J. Math. 114(2) (1984), 399–445.CrossRefGoogle Scholar
Seth, A. and Vaidyanathan, P., ‘AF-algebras and rational homotopy theory’, New York J. Math. 26 (2020), 931–949.Google Scholar
Seth, A. and Vaidyanathan, P., ‘ $K$ -stability of continuous $C(X)$ -algebras’, Proc. Amer. Math. Soc. 148(9) (2020), 3897–3909.CrossRefGoogle Scholar
Strom, J., Modern Classical Homotopy Theory, Graduate Studies in Mathematics, 127 (American Mathematical Society, Providence, RI, 2011).CrossRefGoogle Scholar
Thom, R., L’homologie des espaces fonctionnels (Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège) (Masson, Paris, 1957).Google Scholar
Thomsen, K., ‘Nonstable $K$ -theory for operator algebras’, K-Theory 4(3) (1991), 245–267.CrossRefGoogle Scholar
Vaidyanathan, P., ‘Rokhlin dimension and equivariant bundles’, J. Operator Theory 87(2) (2022), 487–509.CrossRefGoogle Scholar
Whitehead, G. W., Elements of Homotopy Theory (Springer, New York, 1978).10.1007/978-1-4612-6318-0CrossRefGoogle Scholar