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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$L^{2}-$homology over traced *-algebras

Author: William L. Paschke
Journal: Trans. Amer. Math. Soc. 349 (1997), 2229-2251
MSC (1991): Primary 46M20, 16E30, 17B37
MathSciNet review: 1407708
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Abstract: Given a unital complex *-algebra $A$, a tracial positive linear functional $\tau $ on $A$ that factors through a *-representation of $A$ on Hilbert space, and an $A$-module $M$ possessing a resolution by finitely generated projective $A$-modules, we construct homology spaces $H_k(A,\tau ,M)$ for $k = 0, 1, \ldots $. Each is a Hilbert space equipped with a *-representation of $A$, independent (up to unitary equivalence) of the given resolution of $M$. A short exact sequence of $A$-modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for $A$-invariant subspaces of $L^2(A,\tau )^n$ gives well-behaved Betti numbers and an Euler characteristic for $M$ with respect to $A$ and $\tau $.

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  • 1. D. Arapura, P. Bressler, M. Ramachandran, On the fundamental group of a compact Kähler manifold, Duke Math. J. 68 (1992), 477 - 488.MR 94e:57040
  • 2. M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32-3 (1976), 43 - 72. MR 54:8741
  • 3. M. E. B. Bekka, A. Valette, Group cohomology, harmonic functions and the first $L^2$-Betti number, preprint, 1994.
  • 4. K. S. Brown, Cohomology of groups, Springer, 1982. MR 83k:20002
  • 5. J. Cheeger, M. Gromov, Bounds on the von Neumann dimension of $L_2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Diff. Geometry 21 (1985), 1 - 34. MR 87d:58136
  • 6. J. Cheeger, M. Gromov, $L_2$-cohomology and group cohomology, Topology 25 (1986), 189 - 216. MR 87i:58161
  • 7. J. Cohen, Von Neumann dimension and the homology of covering spaces, Quart. J. Math. Oxford Ser. (2) 30 (1979), 133 - 142. MR 81i:20060
  • 8. J. Dodziuk, De Rham-Hodge theory for $L^2$ cohomology of infinite coverings, Topology 16 (1977), 157 - 165. MR 56:3898
  • 9. N. Jing, J. Zhang, Quantum Weyl algebras and deformations of $U(G)$, Pacific J. Math. 171 (1995), 437 - 454. MR 96k:16069
  • 10. R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras, v. ii, Academic Press, 1986. MR 88d:46106
  • 11. W. L. Paschke, A numerical invariant for finitely generated groups via actions on graphs, Math. Scand. 72 (1993), 148 - 160. MR 94e:20005
  • 12. W. L. Paschke, An invariant for finitely presented CG-modules, Math. Ann. 301 (1995),
    325 - 337. MR 96k:20009
  • 13. J. Rosenberg, Algebraic K-theory and its applications, Springer, 1994. MR 95e:19001
  • 14. C. A. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994. MR 95f:18001

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Additional Information

William L. Paschke
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142

Received by editor(s): September 20, 1995
Article copyright: © Copyright 1997 American Mathematical Society

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