Commutants of von Neumann Modules, Representations of B(E) and Other Topics Related to Product Systems of Hilbert Modules

We review some of our results from the theory of product systems of Hilbert modules [BS00, BBLS00, Ske00a, Ske01, Ske02, Ske03]. We explain that the product systems obtained from a CP-semigroup in [BS00] and in [MS02] are commutants of each other. Then we use this new commutant technique to construct product systems from E0–semigroups on Ba(E) where E is a strongly full von Neumann module. (This improves the construction from [Ske02] for Hilbert modules where existence of a unit vector is required.) Finally, we point out that the Arveson system of a CP-semigroup constructed by Powers from two spatial E0–semigroups is the product of the corresponding spatial Arveson systems as defined (for Hilbert modules) in [Ske01]. It need not coincide with the tensor product of Arveson systems. B(G) (G some Hilbert space) are in many respects the simplest von Neumann algebras. They are factors which contain their commutant, they contain all finite rank operators and this determines completely the theory of all normal representations. Hilbert modules (or, more precisely, von Neumann modules; see Section 2) E over B(G) share this simplicity. E is always isomorphic to B(G,H) (H some other Hilbert space) equipped with the natural right module action and inner product 〈x, y〉 = xy. The algebra B(E) of all adjointable mappings on E is just B(H). If E is a two-sided B(G)–module, i.e. if H carries a normal representation of B(G), then H = G ⊗ H (H some other Hilbert space) and elements b ∈ B(G) act in the natural way as b⊗ idH. The Hilbert space H can be identified naturally with the space {x ∈ E : bx = xb (b ∈ B(G))} of all elements in E which intertwine left and right action of B(G), where h ∈ H corresponds to the intertwiner idG⊗h : g 7→ g ⊗ h. (It is not difficult to see that every intertwiner arises in that way, and that 〈x, y〉 ∈ B(G) = C1 gives the correct scalar product as multiple of 1.) We observe that every von Neumann B(G)–B(G)–module E = B(G) ⊗̄ H ( ⊗̄ stands for strong closure in B(G,G ⊗ H)) contains its commutant with respect to B(G) and is generated by it (in the strong topology) as a right module. 2000 Mathematics Subject Classification. Primary 46L55 60J25; Secondary 46L08 46L57 81S25. This work is supported by a PPP-project by DAAD and DST. c ©0000 (copyright holder)