40 Noncommutative Mathematics for Quantum Systems
If a generating functionalLsatisfies one of these conditions, then
we call it and also the associated Levy process ́ quadraticorGaussian.
The proofs of the preceding two propositions can be carried out
as an exercise or found in [Sch93, Section 5.1].
Proposition 1.5.14 Let L be a generating functional onB. Then the
following are equivalent.
(i) There exists a stateφ:B →Cand a real numberλ> 0 such that
L(b) =λ(
φ(b)−ε(b))for all b∈B.(ii) There exists a Sch ̈urmann triple(ρ,η,L)containing L, in which the
cocycleηis trivial, that is, of the form
η(b) =(
ρ(b)−ε(b))
ω, for all b∈B,for some non-zero vectorω∈D. In this case we will also callηthe
coboundaryof the vectorω.If a generating functionalLsatisfies one of these conditions, then
we call it aPoisson generating functionaland the associated Levy ́
process acompound Poisson process.
Proof To show that (ii) implies (i), setφ(b) = 〈ω〈,ωρ(,ωb)〉ω〉andλ=
‖ω‖^2.
For the converse, let(D,ρ,ω)be the GNS triple for(B,φ)and
check that(ρ,η,L)withη(b) =
(
ρ(b)−ε(b))
ω,b ∈ Bdefines a
Schurmann triple. ̈
Remark 1.5.15 The Schurmann triple for a Poisson generating ̈
functionalL=λ(φ−ε)obtained by the GNS construction forφis
not necessarily surjective. Consider, e.g., a classical additive
R-valued compound Poisson process, whose L ́evy measureμis
not supported on a finite set. Then the construction of a surjective
Schurmann triple in the proof of Theorem 1.5.10 gives the ̈
pre-Hilbert spaceD 0 =span{xk|k=1, 2,.. .} ⊆L^2 (R,μ). On the
other hand, the GNS construction forφleads to the pre-Hilbert
spaceD=span{xk|k=0, 1, 2,.. .}⊆L^2 (R,μ). The cocycleηis the
coboundary of the constant function, which is not contained inD 0.