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Proof. For any Σ-measurable g : X → C with g ≥ 0, we know that a sequence of simple Σ-measurable functions sn exist such that 0 ≤ s1 ≤ s2 ≤ ... 17]). So |sn (x) − g(x)|2 → 0 for all x ∈ X, while of course sn ∈ B∞ (Σ), and so [sn ] ∈ G, for all n. 34] that [sn ] − [g] 2 = |sn − g|2 dµ → 0dµ = 0 which means that [g] is contained in the closure of G in L2 (µ). 14(b)]). Note that [u+ ], [u− ], [v + ], [v − ] ∈ L2 (µ), for example |u+ | = u+ ≤ u+ + u− = |u| ≤ |g| where u = u+ − u− . Since [u+ ], [u− ], [v + ], [v − ] are then contained in G’s closure, so is [g] = [u+ ] − [u− ] + i[v + ] − i[v − ].

Proof. 1). To prove (ii), consider any x ∈ S and y ∈ T. From the Mean Ergodic Theorem it follows that n−1 1 x, U k y → x, P y n k=0 as n → ∞. 3) we see that x, P y = x, Ω Ω, y = x, (Ω ⊗ Ω)y . Since the linear span of S is dense in H, this implies that P y = (Ω ⊗ Ω)y. 3(ii). 6. 4 over to ∗-dynamical systems using cyclic representations. 1 Theorem. Let (A, ϕ, τ ) be a ∗-dynamical system, and consider any A ∈ A and ε > 0. Then the set E = k ∈ N : ϕ A∗ τ k (A) is relatively dense in N. > |ϕ(A)|2 − ε 52 CHAPTER 2.

Here normal refers to the form Tr(ρ·) of the state, where ρ is a density operator, while faithful means that ω(A∗A) > 0 if A = 0. 2. Let M := π(L(G)) and H := π(G), then we prove that (M, H, H) is a bounded quantum system. 24]. 6]. 7]), and since M is ∗-isomorphic to L(G), this means that the elements of M which commute with M are also just the multiples of unity, that is to say M ∩ M′ = C. Since π is injective and π(1) = 1, we can therefore define a trace M → M ∩ M′ (in the sense described above) by tr(π(A)) := tr(A), where tr on the right is the (normalized) trace of L(G).

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