By Volkmar Liebscher

In a chain of papers Tsirelson made out of degree forms of random units or (generalised) random tactics a brand new diversity of examples for non-stop tensor product structures of Hilbert areas brought by way of Arveson for classifying E0-semigroups upto cocyle conjugacy. This paper begins from developing the speak. So the writer connects every one non-stop tensor product method of Hilbert areas with degree kinds of distributions of random (closed) units in zero 1 or R. those degree forms are desk bound and factorise over disjoint periods. In a distinct case of this building, the corresponding degree variety is an invariant of the product method. This exhibits, finishing in a extra systematic method the Tsirelson examples, that the class scheme for product platforms into varieties In, IIn and III isn't really whole. furthermore, in response to a close learn of this sort of degree varieties, the writer constructs for every desk bound factorising degree kind a continual tensor product procedure of Hilbert areas such that this degree kind arises because the sooner than pointed out invariant

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Extra resources for Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces

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11). s. It is a standard procedure now to construct a version of this disintegrating kernel which fulfils the support condition everywhere. 28. We consider the Poisson process Π = e−1 F1 . 10) we find Π (dZ1 ) Π (dZ2 )g(Z1 ∪ Z2 ) f (Z1 , Z2 ) = e−2 F1 (dZ1 ) = e−2 F1 (dZ)g(Z) = e−2 F1 ∗ F1 (dZ)g(Z)2−#Z = F1 (dZ2 )g(Z1 ∪ Z2 ) f (Z1 , Z2 ) ∑ Z1 ∪Z2 =Z,Z1 ∩Z2 =0/ Π ∗ Π (dZ)g(Z)2−#Z f (Z1 , Z2 ) ∑ Z1 ∪Z2 =Z,Z1 ∩Z2 =0/ ∑ Z1 ∪Z2 =Z,Z1 ∩Z2 =0/ f (Z1 , Z2 ) f (Z1 , Z2 ) This shows that qΠ (Z, . ) is the uniform distribution on the finite set { (Z1 , Z2 ) : Z1 ∪ Z2 = Z, Z1 ∩ Z2 = 0/ } .

5. Tensor Products (I) One main operation on product systems is the tensor product. , if E , E are product systems then E ⊗ E = (Et ⊗ Et )t≥0 , Vs,t ⊗Vs,t s,t≥0 is again a product system. It is interesting to ask, what implications this operation has in terms of the measure types associated with E and E . 4 below, that all units of E ⊗E have the form u ⊗ u . Then PEs,t⊗E ,u⊗u = PEs,t,u ⊗ PEs,t ,u , ((s,t) ∈ I0,1 ). Further, this result implies also PEs,t⊗E ,U = PEs,t,U ⊗ PEs,t ,U, ((s,t) ∈ I0,1 ).

7. Let M be a stationary factorising measure type on FT different from { δT }. Then the product system E = E M has at least one unit, corresponding to (ut )µ = µ({ 0/ })−1/2 χ{ 0} / , (t ∈ [0, 1], µ ∈ M0,t ). The product system E U generated by all units of E M is given by EtU = ψ ∈ L2 (M0,t ) : χ{Z:#Z=∞} ψµ = 0 for all µ ∈ µ0,t , (t ∈ R+ ). ˇ ConseMoreover, M E ,u = M and M E ,U = M ◦ l −1 , denoting l the map Z → Z. s. Zˇ = Z and M = { δ0/ } otherwise R ANDOM S ETS AND I NVARIANTS 19 In the sequel, the symbol FKf denotes the set of finite sets: FKf = { Z ⊆ K : #Z < ∞ } .

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