Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-08-01T08:42:55.441Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly torsion generated groups from K-theory of real C*-algebras

Published online by Cambridge University Press:  12 May 2008

A. J. Berrick  and
M. Matthey
A. J. Berrick
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore, berrick@math.nus.edu.sg.
M. Matthey
Affiliation:
Late of: University of Lausanne, IGAT (Institute for Geometry, Algebra and Topology), Bâtiment BCH, EPFL, CH-1015 Lausanne, Switzerland
Get access

Abstract

We pursue the program initiated in [7], which consists of an attempt by means of K-theory to construct a strongly torsion generated group with prescribed center and integral homology in dimensions two and higher. Using algebraic and topological K-theory for real C*-algebras, we realize such a construction up to homological dimension five. We also explore the limits of the K-theoretic approach.

Keywords

Strongly torsion generated groupshomologyK-theoryC*-algebras
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 2 , April 2009 , pp. 309 - 326
Copyright
Copyright © ISOPP 2009

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.)

References

1.Baues, H.-J. and Goerss, P.. A homotopy operation spectral sequence for computation of homotopy groups, Topology 39 (2000), 161–192CrossRefGoogle Scholar
2.Baumslag, G., Dyer, E. and Heller, A.. The topology of discrete groups, J. Pure Appl. Alg. 16 (1980), 1–47CrossRefGoogle Scholar
3.Berrick, A. J.. Torsion generators for all abelian groups, J. Algebra 139 (1991), 190–194CrossRefGoogle Scholar
4.Berrick, A. J.. The plus-construction as a localization, Algebraic K-Theory and its Applications, Procs Trieste 1997, World Scientific, (Singapore, 1999), 313–336Google Scholar
5.Berrick, A. J.. Coefficients for homological functors, in preparationGoogle Scholar
6.Berrick, A. J. and Dwyer, W. G.. The spaces that define algebraic K-theory, Topology 39 (2000), 225–228CrossRefGoogle Scholar
7.Berrick, A. J. and Matthey, M.. Homological realization of prescribed abelian groups via K-theory, Math. Proc. Camb. Phil. Soc. 142 (2007), 249–258CrossRefGoogle Scholar
8.Berrick, A. J. and Matthey, M.. Stable classical groups and strongly torsion generated groups, preprint, 2005Google Scholar
9.Berrick, A. J. and Miller, C. F. III. Strongly torsion generated groups, Math. Proc. Camb. Phil. Soc. 111 1992, 219–229CrossRefGoogle Scholar
10.Boersema, J. L.. Real C*-algebras, united K-theory, and the Künneth formula, K-Theory 26 (2002), 345–402CrossRefGoogle Scholar
11.Boersema, J. L.. The range of united K-theory, J. Functional Analysis 235 (2006), 702–718CrossRefGoogle Scholar
12.Bousfield, A. K.. A classification of K-local spectra, J. Pure Appl. Algebra 66 (1990), 121–163CrossRefGoogle Scholar
13.Claborn, L.. Every abelian group is a class group, Pacific J. Math. 118 (1966), 219–222CrossRefGoogle Scholar
14.Hewitt, B. J.. On the homotopical classification of KO-module spectra, Ph.D. Dissertation, Univ. Illinois Chicago (1996)Google Scholar
15.Higson, N.. Algebraic K-theory of stable C*-algebras, Adv. in Math. 67 (1988), 1–140CrossRefGoogle Scholar
16.Karoubi, M.. K-Theory. An introduction. Springer-Verlag, Grundlehren der math. Wissen. 226, 1978Google Scholar
17.Karoubi, M.. K-théorie algébrique de certaines algèbres d'opérateurs, Lecture Notes in Math. 725, Springer (Berlin, 1979), 254–290Google Scholar
18.Karoubi, M.. Homologie des groupes discrets associés à des algèbres d'opérateurs, J. Operator Theory 15 (1986), 109–161Google Scholar
19.Karoubi, M.. Bott periodicity in topological, algebraic and Hermitian K-theory, in Handbook of K-Theory, ed. Friedlander, E. M., Grayson, D. R., Springer (Berlin, 2005)Google Scholar
20.Leedham-Green, C. R.. The class group of Dedekind domains, Trans. Amer. Math. Soc. 163 (1972), 493–500CrossRefGoogle Scholar
21.Mac Lane, S.. Categories for the Working Mathematician. Graduate Texts in Math. 5, Springer Verlag (Berlin, 1971)Google Scholar
22.Milnor, J.. Introduction to Algebraic K-Theory. Annals of Math. Studies 72, Princeton Univ. Press (Princeton, 1971)Google Scholar
23.Murphy, G. J.. C*-Algebras and Operator Theory. Academic Press, 1990Google Scholar
24.Rørdam, M., Larsen, F. and Laustsen, N.. An Introduction to K-Theory for C*-Algebras. Cambridge Univ. Press, London Math. Soc. Student Texts 49, 2000Google Scholar
25.Rosenberg, J.. Algebraic K-Theory and Its Applications. Graduate Texts in Math. 147, Springer (Berlin, 1994)Google Scholar
26.Rosenberg, J.. The algebraic K-theory of operator algebras, K-Theory 12 (1997), 75–99CrossRefGoogle Scholar
27.Schröder, H.. K-Theory for Real C*-Algebras and Applications. Pitman Advanced Publishing Program, Research Notes in Math. 290, (1993)Google Scholar
28.Suslin, A. A. and Wodzicki, M.. Excision in algebraic K-theory and Karoubi's conjecture, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), (9582–9584)CrossRefGoogle ScholarPubMed
29.Suslin, A. A. and Wodzicki, M.. Excision in algebraic K-theory, Ann. of Math. (2) 136 (1992), 51–122CrossRefGoogle Scholar
30.Switzer, R. M.. Algebraic Topology – Homotopy and Homology. Die Grundlehren der math. Wissen. 212, Springer (Berlin, 1975)Google Scholar
31.Taylor, J. L.. Banach algebras and topology, In Algebras Anal., Proc. Instr. Conf. Birmingham, 1973, (1975), 118–186Google Scholar
32.Wegge-Olsen, N. E.. K-Theory and C*-Algebras. A friendly approach. Oxford Science Publications, Oxford Univ. Press (Oxford, 1993)CrossRefGoogle Scholar
33.Whitehead, J. H. C.. The homotopy type of a special kind of polyhedron, Annales de la Soc. Polon. de Math. 21 (1948), 176–186Google Scholar
34.Whitehead, J. H. C.. A certain exact sequence, Ann. of Math. 52 (1950), 51–110CrossRefGoogle Scholar
35.Wood, R.. Banach algebras and Bott periodicity, Topology 4 (1966), 371–389CrossRefGoogle Scholar