By Heinrich W Guggenheimer

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Proof: The first part of the theorem follows from the fact that G r ([ρ]h ) → ( Z + J )([ρ]h ) as r ↑ 1, for all [ρ]h ∈ Dom( Z ). For minimality, it is required to observe that for λ > 0, (λ − A )−1 − (λ − G r )−1 = (λ − A )−1 ( A − G r ) (λ − G r )−1 ≥ 0, since the restriction of A to the range of (λ − G r )−1 , i. e. to Dom(G r ), is the same as Z + J ≥ G r , and (λ − A )−1 , (λ − G r )−1 are positive. We complete the proof by noting that T∗,t = s − limn→∞ (n/t)n (n/t − A )−n (r ) ≥ s − limn→∞ (n/t)n (n/t − G r )−n = T∗,t for all r, t.

Since by our choice X ≥ 0 as an element of A ⊗ Mn , it is clear that positivity of Tn for each n is equivalent to ✷ the positive definiteness of K T . 9 (Stinespring’s theorem) A linear map T : A → B(H) is CP if and only if there is a triple (K, π, V ) consisting of a Hilbert space K, a unital ∗-homomorphism π : A → B(K) and V ∈ B(H, K) such that T (x) = V ∗ π(x)V for all x ∈ A, and {π(x)V u : u ∈ h, x ∈ A} is total in K. Such a triple, to be called the ‘Stinespring triple’ associated with T , is unique in the sense that if (K , π , V ) is another such triple, then there is a unitary operator : K → K such that π (x) = π(x) ∗ and V = V .

Then the symmetric (or Boson) and antisymmetric (or Fermion) Fock spaces over H, denoted respectively by s (H) and a (H), are defined as s s (H) = ⊕∞ n=0 Hn , a a (H) = ⊕∞ n=0 Hn . We shall be mostly concerned with the symmetric Fock spaces in the present work, and hence for simplicity of notation, we shall use the notation (H) for the symmetric Fock space. Let us mention the basic factorization property of (H). ) 0 u ⊗ = 1. ) is the minimal Kolmogorov decomposition for the positive definite kernel H×H → C given by u, v → exp( u, v ).

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