New PDF release: Reviews in Mathematical Physics - Volume 9

By H. Araki, V. Bach, J. Yngvason (Editors)

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In the finite-dimensional case Axioms AHS1 and AHS2 are sufficient to construct a C ∗ -algebra AC whose state space is K; as a Banach space A = A(K) with the sup-norm. To cover the general case, more axioms are needed. Axiom AHS3. The σ-convex hull of ∂e K is a split face of K. Here the σ-convex hull in question consists of all sums i λi ρi , where ρi ∈ ∂e K, λi ∈ [0, 1], i λi = 1, and the sum converging in the norm topology (regarding K as a subset of the dual of the Banach space A(K)). A face F of K is split if there exists another face F such that K = F ⊕c F (direct convex sum).

Hence p(τ, τλ ) → 1. Now take an arbitrary atom αλ ∈ τλ⊥ ∧ F , and complete to a basis {αλ , αλ , τλ }, where αλ ∈ ρλ ∨ σλ . Again, the definition of a basis implies that p(αλ , τ ) → 0. Hence by our second definition of convergence ρλ ∨ σλ → ρ ∨ σ. Secondly, we show that Qλ → Q and Rλ → R, where Q and R are twodimensional subspaces of F , implies Qλ ∧ Rλ → Q ∧ R (we assume Q = R, so eventually Qλ = Rλ ). Let α = Q⊥ ∧F , β = R⊥ ∧F , γ = Q∧R, and γλ = Qλ ∧Rλ ; as a simple dimension count shows, these are all atoms.

1) i=1 here {ei } is a basis of Q; it is easily seen that pQ is independent of the choice of this basis (cf. [59]). Definition 7. Let P be a well-behaved transition probability space. 2) j where λj ∈ R, and {Qj } is an orthogonal family of orthoclosed subsets of P (cf. 1)) for which j pQj equals the unit function on P. Proposition 2. 16)) then any f ∈ A00 (P) has a unique spectral resolution. By the previous section this applies, in particular, to a transition probability space P satisfying Axioms 1 and 2.

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Reviews in Mathematical Physics - Volume 9 by H. Araki, V. Bach, J. Yngvason (Editors)


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