Hilbert modules and modules over finite von Neumann algebras and applications to L2-invariants

Throughout this paper A is a finite von Neumann algebra and tr :A −→ C is a finite normal faithful trace. Recall that a von Neumann algebra is finite if and only if it possesses such a trace. Let l2(A) be the Hilbert space completion of A which is viewed as a pre-Hilbert space by the inner product 〈a, b〉 = tr(ab∗). A finitely generated Hilbert A-module V is a Hilbert space V together with a left operation of A by C-linear maps such that λ · 1A acts by scalar multiplication with λ on V for λ ∈ C and there exists a unitary A-embedding of V in l2(A)n = ⊕i=1l(A). In the sequel A operates always from the left unless explicitly stated differently. A morphism of finitely generated HilbertA-modules is a boundedA-operator. Let {fin. gen. Hilb. A-mod.} be the category of finitely generated Hilbert A-modules. This category plays an important role in the construction of L2-invariants of finite connected CW -complexes such as L2-Betti numbers and Novikov-Shubin invariants. For a survey on L2-(co)homology we refer for instance to [18], [32], [44]. More information about L2-invariants can be found for instance in [1], [5], [8], [9] [13], [14], [17], [19], [24], [25] [26], [29], [30], [31], [34], [40], [43]. These constructions of L2-invariants use the rich functional analytic structure. However, it is a consequence of standard facts about von Neumann algebras that they can be interpreted purely algebraically. Namely, we will prove (see Theorem 2.2) if {fin. gen. proj. A-mod.} denotes the category of finitely generated projective A-modules