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The Toeplitz operator proof of noncommutative Bott periodicity

Part of: Selfadjoint operator algebras $K$-theory and operator algebras Topological (rings and) algebras with an involution

Published online by Cambridge University Press:  09 April 2009

N. Christopher Phillips
N. Christopher Phillips
Affiliation:
University of GeorgiaAthens, GA 30602, USA
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Abstract

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We adapt the Toeplitz operator proof of Bott periodicity to give a short direct proof of Bott periodicity for the representable K-theory of σ-C*-algebras. We further show how the use of this proof and the right definitions simplifies the derivation of the basic properties of representable K-theory.

MSC classification

Secondary: 19K99: None of the above, but in this section 46K99: None of the above, but in this section 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 2 , October 1992 , pp. 229 - 251
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Atiyah, M. F., ‘Bott periodicity and the index of elliptic operators’, Quart. J. Math. Oxford Ser. 2, 19 (1968), 113–140.CrossRefGoogle Scholar
[2]Atiyah, M. F., ‘Algebraic topology and operators in Hilbert space’, in: Lectures in modern analysis I (ed. Taam, C. T.), Lecture Notes in Math. 103 (Springer, Berlin, 1969), pp. 101–121.Google Scholar
[3]Atiyah, M. F. and Bott, R., ‘On the periodicity theorem for complex vector bundles’, Acta Math. 112 (1964), 229–247.CrossRefGoogle Scholar
[4]Blackadar, B., K-theory for operator algebras, MSRI publications no. 5 (Springer-Verlag, Berlin, 1986).CrossRefGoogle Scholar
[5]Cuntz, J., ‘K-theory and C*-algebras’, in: Algebraic K-theory, number theory, geometry, and analysis, Proceedings, Bielefeld 1982 (ed. Bak, A.), Lecture Notes in Math. 1046, (Springer, Berlin, 1984), pp. 55–79.Google Scholar
[6]Gootman, E. C. and Lazar, A. J., ‘Applications of non-commutative duality to crossed product C*-algebras determined by an action or coaction’, Proc. London Math. Soc. 59 (1989), 593–624.CrossRefGoogle Scholar
[7]Karoubi, M., ‘Espaces classifiants en K-théorie’, Trans. Amer. Math. Soc. 147 (1970), 75–115.Google Scholar
[8]Kasparov, G. G., ‘The operator K-functor and extensions of C*-algebras’, Izv. Akad. Nauk SSSR, Ser. Math. 44 (1980), 571–636 (in Russian);Google Scholar
English translation in Math. USSR Izvestija 16 (1981), 513–572.CrossRefGoogle Scholar
[9]Mingo, J. A., ‘K-theory and multipliers of stable C*-algebras’, Trans. Amer. Math. Soc. 299 (1987), 397–411.Google Scholar
[10]Phillips, N. C., Equivariant K-theory and freeness of group actions on C*-algebras, Lecture Notes in Math. 1274 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[11]Phillips, N. C., ‘Inverse limits of C*-algebras’, J. Operator Theory 19 (1988), 159–195.Google Scholar
[12]Phillips, N. C., ‘Representable K-theory for σ-C*-algebras’, K-Theory 3 (1989), 441–478.CrossRefGoogle Scholar
[13]Phillips, N. C., ‘Classifying algebras for the K-theory of σ-C*-algebras’, Canad. J. Math. 41 (1989), 1021–1089.CrossRefGoogle Scholar
[14]Phillips, N. C., ‘C∞ loop algebras and noncommutative Bott periodicity’, Trans. Amer. Math. Soc. 325 (1991), 631–659.Google Scholar
[15]Phillips, N. C., ‘K-theory and noncommutative homotopy theory’, in: Operator theory, operator algebras and applications (eds. Arveson, W. B. and Douglas, R. G.), Proceedings of Symposia in Pure Mathematics 51, part 2 (Amer. Math. Soc., Providence, 1990), pp. 255–265.CrossRefGoogle Scholar
[16]Phillips, N. C., ‘K-theory for Fréchet algebras’, International J. Math. 2 (1991), 77–129.CrossRefGoogle Scholar
[17]Schochet, C., ‘Topological methods for C*-algebras III: axiomatic homology’, Pacific J. Math. 114 (1984), 399–445.CrossRefGoogle Scholar
[18]Segal, G., ‘The representation ring of a compact Lie group’, Publ. Math. Inst. Hautes Études Sci. 34 (1968), 113–128.CrossRefGoogle Scholar
[19]Taylor, J. L., ‘Banach algebras and topology’, in: Algebras in analysis (Academic Press, London, 1975), pp. 118–186.Google Scholar
[20]Weidner, J., Topological invariants for generalized operator algebras, (Ph.D. Thesis, Heidelberg, 1987).Google Scholar
[21]Weidner, J., ‘KK-groups for generalized operator algebras I’, K-Theory 3 (1989), 57–77.CrossRefGoogle Scholar