𝐿²-homology over traced *-algebras

Paschke, William

doi:10.1090/S0002-9947-97-01910-7

$L^2$-homology over traced *-algebras
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by William L. Paschke PDF

Trans. Amer. Math. Soc. 349 (1997), 2229-2251 Request permission

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 $\tau$.

References

D. Arapura, P. Bressler, and M. Ramachandran, On the fundamental group of a compact Kähler manifold, Duke Math. J. 68 (1992), no. 3, 477–488. MR 1194951, DOI 10.1215/S0012-7094-92-06819-0
M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. MR 0420729
M. E. B. Bekka, A. Valette, Group cohomology, harmonic functions and the first $L^2$-Betti number, preprint, 1994.
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Jeff Cheeger and Mikhael Gromov, Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1–34. MR 806699
Jeff Cheeger and Mikhael Gromov, $L_2$-cohomology and group cohomology, Topology 25 (1986), no. 2, 189–215. MR 837621, DOI 10.1016/0040-9383(86)90039-X
Joel M. Cohen, von Neumann dimension and the homology of covering spaces, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 133–142. MR 534828, DOI 10.1093/qmath/30.2.133
Jozef Dodziuk, de Rham-Hodge theory for $L^{2}$-cohomology of infinite coverings, Topology 16 (1977), no. 2, 157–165. MR 445560, DOI 10.1016/0040-9383(77)90013-1
Naihuan Jing and James Zhang, Quantum Weyl algebras and deformations of $U(g)$, Pacific J. Math. 171 (1995), no. 2, 437–454. MR 1372238, DOI 10.2140/pjm.1995.171.437
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
William L. Paschke, A numerical invariant for finitely generated groups via actions on graphs, Math. Scand. 72 (1993), no. 1, 148–160. MR 1226002, DOI 10.7146/math.scand.a-12442
William L. Paschke, An invariant for finitely presented $\textbf {C}G$-modules, Math. Ann. 301 (1995), no. 2, 325–337. MR 1314590, DOI 10.1007/BF01446632
Jonathan Rosenberg, Algebraic $K$-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994. MR 1282290, DOI 10.1007/978-1-4612-4314-4
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136