\(L^ 2-\)homology over traced *-algebras. (English) Zbl 0878.46053
Summary: 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 \(\tau\).
MSC:
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
| 46K10 | Representations of topological algebras with involution |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
Keywords:
unital complex *-algebra; tracial positive linear functional; *-representation; projective \(A\)-modules; homology spaces; exact sequence; Künneth formula; tensor products; von Neumann dimension; \(A\)-invariant subspaces; Betti numbers; Euler characteristicReferences:
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