Classifying Algebras for the K-Theory of σ-C*-Algebras

Phillips, N. Christopher

Canadian journal of mathematics, Tome 41 (1989) no. 6, pp. 1021-1089

Résumé

In topology, the representable K-theory of a topological space X is defined by the formulas RK0(X) = [X,Z x BU] and RKl(X) = [X, U], where square brackets denote sets of homotopy classes of continuous maps, is the infinite unitary group, and BU is a classifying space for U. (Note that ZxBU is homotopy equivalent to the space of Fredholm operators on a separable infinite-dimensional Hilbert space.) These sets of homotopy classes are made into abelian groups by using the H-group structures on Z x BU and U. In this paper, we give analogous formulas for the representable K-theory for α-C*-algebras defined in [20].

Détail

DOI : 10.4153/CJM-1989-046-2

Bibliographie

[1] 1. Adams, J. F., Stable homotopy and generalized homology, part III, Chicago Lectures in Mathematics (University of Chicago Press, 1974). Google Scholar

[2] 2. Anderson, J., Blackadar, B.and Haagerup, U., Minimal projections in the reduced group C*- algebra ofZn * Zm, J. Operator Theory, to appear. Google Scholar

[3] 3. Blackadar, B., A simple C*-algebra with no nontrivial projections, Proc. Amer. Math. Soc. 78 (1980), 504–508. Google Scholar

[4] 4. Blackadar, B., shape theory for C*-algebras, Math. Scand. 56 (1985), 249–275. Google Scholar

[5] 5. Blackadar, B., K-theory for operator algebras, MSRI publications no. 5 (Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1986). Google Scholar

[6] 6. Blackadar, B., Comparison theory in simple C*-algebras, pages 21–54 in: Operator algebras and applications Volume 1 : Structure theory; K-theory, geometry and topology, London Mathematical Society Lecture Note Series no. 135 (Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, 1988). Google Scholar

[7] 7. Brown, L. G., Ext of certain free product C*- algebras, J. Operator Theory 6 (1981), 135–141. Google Scholar

[8] 8. Cuntz, J., A new look at KK-theory, -Theory / (1987), 31–51. Google Scholar

[9] 9. Cuntz, J.and Higson, N., Kuiper's theorem for Hilbert modules, pages 429–435 in: Operator algebras and mathematical physics, Contemporary Math. 62 (Amer. Math. Soc, 1987). Google Scholar

[10] 10. Effros, E. G. and Kaminker, J., Homotopy continuity and shape theory for C*-algebras, pages 152–180 in: Geometric methods in operator algebras, Pitman Research Notes in Math. 123 (Longman Scientific and Technical, 1986). Google Scholar

[11] 11. Karoubi, M., K-theory: an introduction, Grundlehren der Mathematischen Wissenschaften 226 (Springer-Verlag, Berlin, Heidelberg, New York, 1978). Google Scholar

[12] 12. Kasparov, G. G., The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571–636 (in Russian); English translation in Math. USSR Izvestija 16 (1981), 513–572. Google Scholar

[13] 13. Loring, T. A., The torus and noncommutative topology, Ph.D. Thesis, Berkeley (1986). Google Scholar

[14] 14. MacLane, S., Categories for the working mathematician, Graduate Texts in Math 5 (Springer- Verlag, Berlin, Heidelberg, New York, 1971). Google Scholar

[15] 15. Mingo, J. A., K-theory and multipliers of stable C*-algebras, Trans. Amer. Math. Soc. 299 (1987), 397–411. Google Scholar

[16] 16. Mingo, J. A. and Phillips, W. J., Equivariant triviality theorems for Hubert C*-modules, Proc. Amer. Math. Soc. 91 (1984), 225–230. Google Scholar

[17] 17. Pedersen, G. K., SAW*-algebras and corona C*-algebras, contributions to noncommutative topology, J. Operator Theory 75 (1986), 15–32. Google Scholar

[18] 18. Phillips, N. C., Equivariant K-theory and freeness of group actions on C*-algebras, Lecture Notes in Math. 1274 (Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987). Google Scholar

[19] 19. Phillips, N. C., Inverse limits of C*-algebras, J. Operator Theory 19 (1988), 159–195. Google Scholar

[20] 20. Phillips, N. C., Representable K-theory for cr-C*-algebras, K-Theory, to appear. Google Scholar

[21] 21. Phillips, N. C., Inverse limits of C*-algebras and applications, pages 127–185 in: Operator algebras and applications Volume 1: Structure theory; K-theory, geometry and topology, London Mathematical Society Lecture Note Series 135 (Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, 1988). Google Scholar

[22] 22. Rosenberg, J., The role of K-theory in noncommutative algebraic topology, pages 155–182 in: Operator algebras and K-theory, Contemporary Math. 10 (Amer. Math. Soc, 1982). Google Scholar

[23] 23. Schochet, C., Topological methods for C*-algebras II: geometric resolutions and the Kunneth formula, Pacific J. Math. 98 (1982), 443–458. Google Scholar

[24] 24. Schochet, C., Topological methods for C*-algebras IV: mod p homology, Pacific J. Math. 114 (1984), 447–168. Google Scholar

[25] 25. Spanier, E., Quasitopologies, Duke Math. J. 30 (1963), 1–14. Google Scholar

[26] 26. Spanier, E., Algebraic topology (McGraw-Hill, New York, San Francisco, St. Louis, Toronto, London, Sydney, 1966). Google Scholar

[27] 27. Switzer, R. M., Algebraic topology - homotopy and homology, Grundlehren der Mathematischen Wissenschaften 212 (Springer-Verlag, Berlin, Heidelberg, New York, 1975). Google Scholar

[28] 28. Takesaki, M., Theory of operator algebras I (Springer-Verlag, New York, Heidelberg, Berlin, 1979). Google Scholar

[29] 29. Voiculescu, D., Dual algebraic structures on operator algebras related to free products, J. Operator Theory 17 (1987), 85–98. Google Scholar

[30] 30. Weidner, J., Topological invariants for generalized operator algebras, Ph.D. Thesis, Heidelberg (1987). Google Scholar

[31] 31. Whitehead, G. W., Elements of homotopy theory, Graduate Texts in Math. 61 (Springer-Verlag, New York, Heidelberg, Berlin, 1978). Google Scholar

[32] 32. Zekri, R., A new description of Kasparov's theory of C*-algebra extensions, Ph.D. Thesis, Université d'Aix-Marseille II, Faculté des Sciences de Luminy (1986). Google Scholar