## Duke Mathematical Journal

### The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor

#### Article information

Source
Duke Math. J. Volume 55, Number 2 (1987), 431-474.

Dates
First available: 20 February 2004

http://projecteuclid.org/euclid.dmj/1077306030

Mathematical Reviews number (MathSciNet)
MR894590

Zentralblatt MATH identifier
0644.46051

Digital Object Identifier
doi:10.1215/S0012-7094-87-05524-4

#### Citation

Rosenberg, Jonathan; Schochet, Claude. The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K -functor. Duke Mathematical Journal 55 (1987), no. 2, 431--474. doi:10.1215/S0012-7094-87-05524-4. http://projecteuclid.org/euclid.dmj/1077306030.

#### References

• [1]1 S. Araki and H. Toda, Multiplicative structures in ${\rm mod}\,q$ cohomology theories. I, Osaka J. Math. 2 (1965), 71–115.
• [1]2 S. Araki and H. Toda, Multiplicative structures in ${\rm mod}\sb{q}$ cohomology theories. II, Osaka J. Math. 3 (1966), 81–120.
• [2] M. F. Atiyah, Vector bundles and the Künneth formula, Topology 1 (1962), 245–248.
• [3] L. G. Brown, Operator algebras and algebraic $K$-theory, Bull. Amer. Math. Soc. 81 (1975), no. 6, 1119–1121.
• [4] L. G. Brown, Extensions and the structure of $C\sp*$-algebras, Symposia Mathematica, Vol. XX (Convegno sulle Algebre $C\sp*$ e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria $K$, INDAM, Rome, 1975), Academic Press, London, 1976, pp. 539–566.
• [5] L. G. Brown, The universal coefficient theorem for ${\mathrm Ext}$ and quasi-diagonality, Operator Algebras and Group Representations, I (Neptun, 1980), Monographs and Studies in Math., vol. 17, Pitman, London, 1984, Proc. Internat. Conf. at Neptun, Romania, 1980, pp. 60–64.
• [6] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C\sp*$-algebras and $K$-homology, Ann. of Math. (2) 105 (1977), no. 2, 265–324.
• [7] L. G. Brown, P. Green, and M. A. Rieffel, Stable isomorphism and strong Morita equivalence of $C\sp*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363.
• [8] R. C. Busby, Double centralizers and extensions of $C\sp{\ast}$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99.
• [9] A. Connes, An analogue of the Thom isomorphism for crossed products of a $C\sp{\ast}$-algebra by an action of ${\bf R}$, Adv. in Math. 39 (1981), no. 1, 31–55.
• [10] A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183.
• [11] J. Cuntz, A class of $C\sp{\ast}$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C\sp{\ast}$-algebras, Invent. Math. 63 (1981), no. 1, 25–40.
• [12] J. Cuntz, The internal structure of simple $C^{\ast}$-algebras, Operator Algebras and Applications, Part I (Kingston, Ont., 1980) ed. R. V. Kadison, Proc. Symp. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 85–115.
• [13] J. Cuntz, On the homotopy groups of the space of endomorphisms of a $C^{\ast}$-algebra (with applications to topological Markov chains), Operator Algebras and Group Representations, I (Neptun, 1980), Monographs and Studies in Math., vol. 17, Pitman, London, 1984, Proc. Internat. Conf. at Neptun, Romania, 1980, pp. 124–137.
• [14] J. Cuntz, $K$-theory and $C^{\ast}$-algebras, Algebraic $K$-theory, Number Theory, Geometry and Analysis (Bielefeld, 1982) ed. A. Bak, Lecture Notes in Math., vol. 1046, Springer, Heidelberg and New York, 1984, Proc. Internat. Conf. at Bielefeld, 1982, pp. 55–79.
• [15] R. G. Douglas, $C\sp{\ast}$-algebra extensions and $K$-homology, Annals of Mathematics Studies, vol. 95, Princeton University Press, Princeton, N.J., 1980.
• [16] T. Fack, $K$-théorie bivariante de Kasparov, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 149–166.
• [17] T. Fack and G. Skandalis, Connes' analogue of the Thom isomorphism for the Kasparov groups, Invent. Math. 64 (1981), no. 1, 7–14.
• [18] D. Handelman, Extensions for AF $C\sp{\ast}$ algebras and dimension groups, Trans. Amer. Math. Soc. 271 (1982), no. 2, 537–573.
• [19] D. S. Kahn, J. Kaminker, and C. Schochet, Generalized homology theories on compact metric spaces, Michigan Math. J. 24 (1977), no. 2, 203–224.
• [20] J. Kaminker and C. Schochet, $K$-theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras, Trans. Amer. Math. Soc. 227 (1977), 63–107.
• [21] G. G. Kasparov, The operator $K$-functor and extensions of $C\sp{\ast}$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719, Math. USSR Izv. 16 (1981), 513–572.
• [22] G. G. Kasparov, $K$-theory, group $C^{\ast}$-algebras, and higher signatures (conspectus), preprint, Chernogolovka, 1981.
• [23] G. G. Kasparov, Lorentz groups: $K$-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR 275 (1984), no. 3, 541–545.
• [24] G. G. Kasparov, Operator $K$-theory and its applications: elliptic operators, group representations, higher signatures, $C\sp \ast$-extensions, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 987–1000.
• [25] A. S. Miščenko and A. T. Fomenko, The index of elliptic operators over $C^{\ast}$-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 43 (1979), no. 4, 831–859, 967, Math. USSR Izv. 15 (1980), 87–117.
• [26a] M. Pimsner, S. Popa, and D. Voiculescu, Homogeneous $C^{\ast}$-extensions of $C(X)\otimes \mathcal{K}(H)$. I, J. Operator Theory 1 (1979), no. 1, 55–108.
• [26b] M. Pimsner, S. Popa, and D. Voiculescu, Homogeneous $C^{\ast}$-extensions of $C(X)\otimes K(H)$. II, J. Operator Theory 4 (1980), no. 2, 211–249.
• [27] M. Pimsner and D. Voiculescu, $K$-groups of reduced crossed products by free groups, J. Operator Theory 8 (1982), no. 1, 131–156.
• [28] J. Rosenberg, Homological invariants of extensions of $C^{\ast}$-algebras, Operator Algebras and Applications, Part I (Kingston, Ont., 1980) ed. R. V. Kadison, Proc. Symp. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 35–75.
• [29] J. Rosenberg, The role of $K$-theory in non-commutative algebraic topology, Operator Algebras and $K$-theory (San Francisco, Calif., 1981) eds. R. G. Douglas and C. Schochet, Contemporary Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155–182.
• [30] J. Rosenberg, $C\sp{\ast}$-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. (1983), no. 58, 197–212 (1984).
• [31] J. Rosenberg and C. Schochet, The classification of extensions of $C^{\ast}$-algebras, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 105–110.
• [32] J. Rosenberg and C. Schochet, Comparing functors classifying extensions of $C^{\ast}$-algebras, J. Operator Theory 5 (1981), no. 2, 267–282.
• [33] C. Schochet, Homogeneous extensions of $C^{\ast}$-algebras and $K$-theory I, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 1, part 1, 715–718.
• [34] C. Schochet, Homogeneous extensions of $C^{\ast}$-algebras and $K$-theory II, Amer. J. Math. 105 (1983), no. 3, 595–622.
• [35] C. Schochet, Topological methods for $C^{\ast}$-algebras II: geometry resolutions and the Künneth formula, Pacific J. Math. 98 (1982), no. 2, 443–458.
• [36] C. Schochet, Topological methods for $C^{\ast}$-algebras III: axiomatic homology, Pacific J. Math. 114 (1984), no. 2, 399–445.
• [37] C. Schochet, Topological methods for $C^{\ast}$-algebras IV: mod $p$ homology, Pacific J. Math. 114 (1984), no. 2, 447–468.
• [38] G. Skandalis, On the group of extensions relative to a semifinite factor, J. Operator Theory 13 (1985), no. 2, 255–263.
• [39] G. Skandalis, Exact sequences for the Kasparov groups of graded algebras, Canad. J. Math. 37 (1985), no. 2, 193–216.
• [40] C. F. Bödigheimer, Splitting the Künneth sequence in $K$-theory, Math. Ann. 242 (1979), no. 2, 159–171.
• [41] A. Deutz, The splitting of the Künneth sequence in $K$-theory for $C^{\ast}$-algebras, Ph.D. Dissertation, Wayne State Univ., 1981.