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That is to say λ(F ) < ∞, where λ is the Lebesgue measure on R2n . 2 Remark. If the phase point is confined to a set F ⊂ R2n , then we can view F as the phase space of the system (whether F has finite volume or not), taking the σ-algebra Σ of measurable sets in F as the intersections of the Borel sets of R2n with F . ) We then replace the Lebesgue measure by its restriction to F (assuming F is Lebesgue measurable), and we use probability measures on F , instead of on R2n . Also, the observables will be represented by Σ-measurable functions on F , and the observable algebra will be B∞ (F ) := B∞ (Σ).

3) holds for all A ∈ S∗ and B ∈ T, then (A, ϕ, τ ) is ergodic. Proof. 3 in terms of any cyclic representation of (A, ϕ). Suppose (A, ϕ, τ ) is ergodic. 2. This proves (i). 5(i) using the Cauchy-Schwarz inequality |ϕ(AC)| ≤ A∗ ϕ C ϕ with C = n−1 k 1 k=0 τ (B) − ϕ(B). 4), hold for all A ∈ S∗ and B ∈ T. 4(ii) and the identities ι(A∗ ), Ω = ϕ(A) and Ω, ι(B) = ϕ(B), that P = Ω⊗Ω. 3, confirming (ii). This characterizes ergodicity in terms of mixing. 7 Example. Let A be the unital ∗-algebra of 2 × 2-matrices with entries in C, the involution being the conjugate transpose.

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. RECURRENCE AND ERGODICITY IN ∗-ALGEBRAS Proof. 3 in terms of any cyclic representation of (A, ϕ). Set x = ι(A). 1) it is clear that Ω = ι(1) is a fixed point of U, so Ω, x = P Ω, x = Ω, P x .