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

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

Jonathan Rosenberg and Claude Schochet

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

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Primary Subjects: 46L80

Secondary Subjects: 19K33, 46M20, 58G12

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Mathematical Reviews number (MathSciNet): MR894590
Zentralblatt MATH identifier: 0644.46051
Digital Object Identifier: doi:10.1215/S0012-7094-87-05524-4

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References

[1]¹ S. Araki and H. Toda, Multiplicative structures in ${\rm mod}\,q$ cohomology theories. I, Osaka J. Math. 2 (1965), 71–115.

Mathematical Reviews (MathSciNet): MR32:449

Zentralblatt MATH: 0129.15201

Project Euclid: euclid.ojm/1200691225

[1]² S. Araki and H. Toda, Multiplicative structures in ${\rm mod}\sb{q}$ cohomology theories. II, Osaka J. Math. 3 (1966), 81–120.

Mathematical Reviews (MathSciNet): MR34:2003

Zentralblatt MATH: 0152.22102

Project Euclid: euclid.ojm/1200691578

[2] M. F. Atiyah, Vector bundles and the Künneth formula, Topology 1 (1962), 245–248.

Mathematical Reviews (MathSciNet): MR27:767

Zentralblatt MATH: 0108.17801

Digital Object Identifier: doi:10.1016/0040-9383(62)90107-6

[3] L. G. Brown, Operator algebras and algebraic $K$-theory, Bull. Amer. Math. Soc. 81 (1975), no. 6, 1119–1121.

Mathematical Reviews (MathSciNet): MR52:3971

Zentralblatt MATH: 0332.46038

Digital Object Identifier: doi:10.1090/S0002-9904-1975-13943-7

Project Euclid: euclid.bams/1183537428

[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.

Mathematical Reviews (MathSciNet): MR56:16400

Zentralblatt MATH: 0366.46047

[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.

Mathematical Reviews (MathSciNet): MR85m:46066

Zentralblatt MATH: 0548.46055

[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.

Mathematical Reviews (MathSciNet): MR56:16399

Zentralblatt MATH: 0376.46036

Digital Object Identifier: doi:10.2307/1970999

JSTOR: links.jstor.org

[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.

Mathematical Reviews (MathSciNet): MR57:3866

Zentralblatt MATH: 0362.46043

Project Euclid: euclid.pjm/1102811432

[8] R. C. Busby, Double centralizers and extensions of $C\sp{\ast}$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99.

Mathematical Reviews (MathSciNet): MR37:770

Zentralblatt MATH: 0165.15501

[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.

Mathematical Reviews (MathSciNet): MR82j:46084

Zentralblatt MATH: 0461.46043

Digital Object Identifier: doi:10.1016/0001-8708(81)90056-6

[10] A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183.

Mathematical Reviews (MathSciNet): MR87h:58209

Zentralblatt MATH: 0575.58030

Digital Object Identifier: doi:10.2977/prims/1195180375

[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.

Mathematical Reviews (MathSciNet): MR82f:46073b

Zentralblatt MATH: 0461.46047

Digital Object Identifier: doi:10.1007/BF01389192

[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.

Mathematical Reviews (MathSciNet): MR84h:46072

Zentralblatt MATH: 0502.46039

[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.

Mathematical Reviews (MathSciNet): MR86a:46093

Zentralblatt MATH: 0561.46025

[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.

Mathematical Reviews (MathSciNet): MR86d:46071

Zentralblatt MATH: 0548.46056

Digital Object Identifier: doi:10.1007/BFb0072018

[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.

Mathematical Reviews (MathSciNet): MR82c:46082

Zentralblatt MATH: 0458.46049

[16] T. Fack, $K$-théorie bivariante de Kasparov, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 149–166.

Mathematical Reviews (MathSciNet): MR86f:46075

Zentralblatt MATH: 0542.46039

[17] T. Fack and G. Skandalis, Connes' analogue of the Thom isomorphism for the Kasparov groups, Invent. Math. 64 (1981), no. 1, 7–14.

Mathematical Reviews (MathSciNet): MR82g:46113

Zentralblatt MATH: 0482.46043

Digital Object Identifier: doi:10.1007/BF01393931

[18] D. Handelman, Extensions for AF $C\sp{\ast}$ algebras and dimension groups, Trans. Amer. Math. Soc. 271 (1982), no. 2, 537–573.

Mathematical Reviews (MathSciNet): MR84e:46063

Zentralblatt MATH: 0517.46051

[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.

Mathematical Reviews (MathSciNet): MR57:13921

Zentralblatt MATH: 0384.55001

Digital Object Identifier: doi:10.1307/mmj/1029001885

Project Euclid: euclid.mmj/1029001885

[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.

Mathematical Reviews (MathSciNet): MR55:4138

Zentralblatt MATH: 0368.46054

JSTOR: links.jstor.org

[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.

Mathematical Reviews (MathSciNet): MR81m:58075

Zentralblatt MATH: 0448.46051

[22] G. G. Kasparov, $K$-theory, group $C^{\ast}$-algebras, and higher signatures (conspectus), preprint, Chernogolovka, 1981.

Mathematical Reviews (MathSciNet): MR1388299

Zentralblatt MATH: 0957.58020

Digital Object Identifier: doi:10.1017/CBO9780511662676.007

[23] G. G. Kasparov, Lorentz groups: $K$-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR 275 (1984), no. 3, 541–545.

Mathematical Reviews (MathSciNet): MR85k:22015

Zentralblatt MATH: 0584.22004

[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.

Mathematical Reviews (MathSciNet): MR87c:58119

Zentralblatt MATH: 0571.46047

[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.

Mathematical Reviews (MathSciNet): MR81i:46075

Zentralblatt MATH: 0416.46052

[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.

Mathematical Reviews (MathSciNet): MR82e:46093a

Zentralblatt MATH: 0455.46060

[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.

Mathematical Reviews (MathSciNet): MR82e:46093b

Zentralblatt MATH: 0484.46059

[27] M. Pimsner and D. Voiculescu, $K$-groups of reduced crossed products by free groups, J. Operator Theory 8 (1982), no. 1, 131–156.

Mathematical Reviews (MathSciNet): MR84d:46092

Zentralblatt MATH: 0533.46045

[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.

Mathematical Reviews (MathSciNet): MR85h:46099

Zentralblatt MATH: 0502.46052

[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.

Mathematical Reviews (MathSciNet): MR84h:46097

Zentralblatt MATH: 0486.46052

[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).

Mathematical Reviews (MathSciNet): MR85g:58083

Zentralblatt MATH: 0526.53044

Digital Object Identifier: doi:10.1007/BF02953775

[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.

Mathematical Reviews (MathSciNet): MR82a:46066

Zentralblatt MATH: 0473.46047

Digital Object Identifier: doi:10.1090/S0273-0979-1981-14873-4

Project Euclid: euclid.bams/1183547855

[32] J. Rosenberg and C. Schochet, Comparing functors classifying extensions of $C^{\ast}$-algebras, J. Operator Theory 5 (1981), no. 2, 267–282.

Mathematical Reviews (MathSciNet): MR84j:46101

Zentralblatt MATH: 0472.46050

[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.

Mathematical Reviews (MathSciNet): MR81i:46096

Zentralblatt MATH: 0441.46057

Digital Object Identifier: doi:10.1090/S0273-0979-1980-14803-X

Project Euclid: euclid.bams/1183546475

[34] C. Schochet, Homogeneous extensions of $C^{\ast}$-algebras and $K$-theory II, Amer. J. Math. 105 (1983), no. 3, 595–622.

Mathematical Reviews (MathSciNet): MR85e:46035

Zentralblatt MATH: 0531.46047

Digital Object Identifier: doi:10.2307/2374315

JSTOR: links.jstor.org

[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.

Mathematical Reviews (MathSciNet): MR84g:46105b

Zentralblatt MATH: 0439.46043

Project Euclid: euclid.pjm/1102734267

[36] C. Schochet, Topological methods for $C^{\ast}$-algebras III: axiomatic homology, Pacific J. Math. 114 (1984), no. 2, 399–445.

Mathematical Reviews (MathSciNet): MR86g:46102

Zentralblatt MATH: 0491.46061

Project Euclid: euclid.pjm/1102708717

[37] C. Schochet, Topological methods for $C^{\ast}$-algebras IV: mod $p$ homology, Pacific J. Math. 114 (1984), no. 2, 447–468.

Mathematical Reviews (MathSciNet): MR86g:46103

Zentralblatt MATH: 0491.46062

Project Euclid: euclid.pjm/1102708718

[38] G. Skandalis, On the group of extensions relative to a semifinite factor, J. Operator Theory 13 (1985), no. 2, 255–263.

Mathematical Reviews (MathSciNet): MR86d:46073

Zentralblatt MATH: 0603.46065

[39] G. Skandalis, Exact sequences for the Kasparov groups of graded algebras, Canad. J. Math. 37 (1985), no. 2, 193–216.

Mathematical Reviews (MathSciNet): MR86d:46072

Zentralblatt MATH: 0603.46064

Digital Object Identifier: doi:10.4153/CJM-1985-013-x

[40] C. F. Bödigheimer, Splitting the Künneth sequence in $K$-theory, Math. Ann. 242 (1979), no. 2, 159–171.

Mathematical Reviews (MathSciNet): MR80k:55013

Zentralblatt MATH: 0412.55003

Digital Object Identifier: doi:10.1007/BF01420413

[41] A. Deutz, The splitting of the Künneth sequence in $K$-theory for $C^{\ast}$-algebras, Ph.D. Dissertation, Wayne State Univ., 1981.

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