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Extending cellular cohomology to $ C\sp *$-algebras

Authors: Ruy Exel and Terry A. Loring
Journal: Trans. Amer. Math. Soc. 329 (1992), 141-160
MSC: Primary 46L80; Secondary 19K56, 46M20, 58A10, 58G12
MathSciNet review: 1024770
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Abstract: A filtration on the $ K$-theory of $ {C^*}$-algebras is introduced. The relative quotients define groups $ {H_n}(A),n \geq 0$, for any $ {C^*}$-algebra $ A$, which we call the spherical homology of $ A$. This extends cellular cohomology in the sense that

$\displaystyle {H_n}(C(X)) \otimes {\mathbf{Q}} \cong {H^n}(X;{\mathbf{Q}})$

for $ X$ a finite CW-complex. While no extension of cellular cohomology which is derived from a filtration on $ K$-theory can be additive, Morita-invariant, and continuous, $ {H_n}$ is shown to be infinitely additive, Morita invariant for unital $ {C^*}$-algebras, and continuous in limited cases.

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  • [1] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics. MR 0402720
  • [2] Bruce Blackadar, 𝐾-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867
  • [3] Joachim Cuntz, The 𝐾-groups for free products of 𝐶*-algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 81–84. MR 679696
  • [4] Ruy Exel, Rotation numbers for automorphisms of 𝐶* algebras, Pacific J. Math. 127 (1987), no. 1, 31–89. MR 876017
  • [5] P. de la Harpe and G. Skandalis, Déterminant associé à une trace sur une algébre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 241–260 (French, with English summary). MR 743629
    P. de la Harpe and G. Skandalis, Produits finis de commutateurs dans les 𝐶*-algèbres, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 4, 169–202 (French, with English summary). MR 766279
    Pierre de la Harpe and Georges Skandalis, Sur la simplicité essentielle du groupe des inversibles et du groupe unitaire dans une 𝐶*-algèbre simple, J. Funct. Anal. 62 (1985), no. 3, 354–378 (French, with English summary). MR 794775,
  • [6] Terry A. Loring, The 𝐾-theory of AF embeddings of the rational rotation algebras, 𝐾-Theory 4 (1991), no. 3, 227–243. MR 1106954,
  • [7] Sibe Mardešić and Jack Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR 676973
  • [8] Mihai V. Pimsner, Embedding some transformation group 𝐶*-algebras into AF-algebras, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 613–626. MR 753927,
  • [9] Marc A. Rieffel, 𝐶*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572
  • [10] Jonathan Rosenberg, The role of 𝐾-theory in noncommutative algebraic topology, Operator algebras and 𝐾-theory (San Francisco, Calif., 1981) Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155–182. MR 658514
  • [11] Claude Schochet, Topological methods for 𝐶*-algebras. III. Axiomatic homology, Pacific J. Math. 114 (1984), no. 2, 399–445. MR 757510
  • [12] Graeme Segal, 𝐾-homology theory and algebraic 𝐾-theory, 𝐾-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975) Springer, Berlin, 1977, pp. 113–127. Lecture Notes in Math., Vol. 575. MR 0515311

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Keywords: $ {C^*}$-algebras, homology, $ K$-theory, filtration, determinant
Article copyright: © Copyright 1992 American Mathematical Society