Algebraic K-Theory and its Geometric Applications by Robert M.F. Moss, Charles B. Thomas

By Robert M.F. Moss, Charles B. Thomas

Shipped from united kingdom, please permit 10 to 21 enterprise days for arrival. Algebraic K-Theory and its Geometric functions, paperback, Lecture Notes in arithmetic 108. 86pp. 25cm. ex. lib.

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7, there exists T ∈ B(h ⊕ K) such that W (π(a)) = T π(a) − P Q π(a)T . Writing T = with respect to the canonical decomposition R S of B(h ⊕ K), we obtain α(a) = Ra − ρ(a)R. Consider the map (x) ≡ L(x) − R ∗ ρ(x)R. 7 and also using the fact that is adjoint-preserving, we obtain a self-adjoint H in B(h) such that (x) − 12 ( (1)x + x (1)) = i[H, x]. The proof that R can be chosen from the ultra-weak closure of Sp{α(x)y : x, y ∈ A} and H can be chosen from A is omitted; referring the reader to the original paper by Christensen and Evans [24].

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 . Furthermore, if A is a von Neumann algebra and T is normal, π can be chosen to be normal. 8. For x ∈ A, define a map π(x) on the linear span of vectors λ(y)u by setting π(x)(λ(y)u) = λ(x y)u, and by extending linearly. The complete positivity of T enables us to verify that indeed π(x) is well defined and one has 2 n λ(xi )u i π(x) i=1 2 n ≤ x λ(xi )u i 2 i=1 for any finite collection x1 , .

I, j=1 This completes the proof that (C) extends to a contraction, since the linear span of exponential vectors is a dense subset of (H). It is straightforward to see that (C) is an isometry (respectively unitary) whenever C is so. ✷ For u ∈ H and unitary operator U in H, we define the Weyl operators W (u, U ) by setting W (u, U )e(v) = exp − 1 u 2 2 − u, U v e(u + U v). It is known that the von Neumann algebra generated by the family {W (u, I ) : u ∈ S} is the whole of B( (H)) whenever S is a dense subspace of H.

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