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Transactions of the American Mathematical Society
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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Additional Information

William L. Paschke
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142
Email: paschke@kuhub.cc.ukans.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01910-7
PII: S 0002-9947(97)01910-7
Received by editor(s): September 20, 1995
Article copyright: © Copyright 1997 American Mathematical Society