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Frigerio, A. : Quantum probability and Applications. Lecture Notes in Math. , von Waldenfels: Quantum probability and Applications III. Lecture Notes in Math. : Mathematical Methods of Classical Mechanics. : Extended quantum mechanics. : Topology and Analysis. The Atiyah-Singer Index Formula and GaugeTheoretic Physics. : Int´egration. : Vari´et´es diff´erentielles et analytiques. Fascicule de r´esultats. Paragraphes 1 a` 7. : Groupes et alg`ebres de Lie. : Operator Algebras and Quantum Statistical Mechanics I.
G 50 A. S. Ratiu Next, we show that R = 1. Since R is a projector we have R ≥ 1. To prove equality, note first that if ρ ≥ 0, that is, ψ|ρ|ψ ≥ 0 for all |ψ ∈ M, then we also have ψ|U (g)ρU (g)∗ |ψ ≥ 0 for all |ψ ∈ M and integration over G yields R(ρ) ≥ 0. Thus we showed that ρ ≥ 0 implies R(ρ) ≥ 0. Continuity of the trace in the · 1 –norm gives then for ρ ≥ 0, R(ρ) 1 tr U (g)ρU (g)∗ dµ(g) = = tr R(ρ) = G tr ρ dµ(g) = tr ρ = ρ 1 , G which proves that R = 1. Finally, we need to show that im R ∗ is a Banach Lie subalgebra of L∞ (M).
If v ∈ M, then R ∗ (x)v 2 2 = pα xpα = α∈I pα xpα v 2 ≤ x α∈I 2 pα v 2 = x 2 v 2 α∈I which shows that R ∗ (x) ≤ 1. 7) we have R ∗ 2 (x) = pα xpα pβ = pβ β∈I α∈I pα xpα = R ∗ (x); δαβ pα xpα δαβ = β∈I α∈I α∈I in this computation the σ -continuity of left and right multiplication with an element pβ was used. Thus R ∗ 2 = R ∗ and R ∗ = 1. For any b ∈ m∗ , there is an element ρ ∈ L1 (M) such that x, b = tr(xρ). Thus R ∗ (x), b = tr(R ∗ (x)ρ) = tr(pα xpα ρ) = α∈I tr(xpα ρpα ) = tr x α∈I pα xpα . α∈I σ σ We want to check that xi → x implies that R ∗ (xi ) → R ∗ (x).