By Sergey Neshveyev
The publication addresses mathematicians and physicists, together with graduate scholars, who're attracted to quantum dynamical structures and purposes of operator algebras and ergodic concept. it's the simply monograph in this subject. even though the authors suppose a simple wisdom of operator algebras, they provide targeted definitions of the notions and typically whole proofs of the consequences that are used.
Read or Download Dynamical Entropy in Operator Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge PDF
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Extra info for Dynamical Entropy in Operator Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
N ; ξ1 , . . , ξn ) = Hµ (∨k ξk ) − Hµ (ξk ) + k Hλ (γk ; ξk ). 12) k Any decomposition ϕ = i ϕi deﬁnes a canonical coupling in an obvious way: take the measure on X = S(A) to be ˆi is the ˆi , where δϕ i ϕi (1)δϕ δ-measure concentrated at the point ϕˆi . To put it diﬀerently, given a coupling λ of (A, ϕ) with (Y, µ), and a ﬁnite measurable partition ξ of Y , we can construct a map f : Y → X which sends an atom Z of ξ to the state ϕZ = µ(Z)−1 λ(·⊗1Z ). Consider the image µ = f∗ (µ) of the measure µ, and put λ = λ◦(idA ⊗f ∗ ), where f ∗ : L∞ (X, µ ) → L∞ (Y, µ) is the map induced by f .
N ). Thus by deﬁnition S(λ(γk (·) ⊗ 1Z ), ϕ ◦ γk ). Hλ (γ1 , . . , γn ; ξ1 , . . 2), the right hand side can be written as Hµ (∨k ξk )− µ(Z)S(µ(Z)−1 λ(γk (·)⊗1Z ), ϕ◦γk ). 3) now becomes µ(Z)S(ϕZ ◦ γk )⎠ . 11) Z∈ξk 44 3 Dynamical Entropy Hλ (γ1 , . . , γn ; ξ1 , . . , ξn ) = Hµ (∨k ξk ) − Hµ (ξk ) + k Hλ (γk ; ξk ). 12) k Any decomposition ϕ = i ϕi deﬁnes a canonical coupling in an obvious way: take the measure on X = S(A) to be ˆi is the ˆi , where δϕ i ϕi (1)δϕ δ-measure concentrated at the point ϕˆi .
5 it follows that S(ϕ, ψ) = limi S(ϕ ◦ γi , ψ ◦ γi ) for any ϕ and ψ. This equality could be used as a deﬁnition of relative entropy for nuclear algebras, which would in fact be suﬃcient for us in most cases. 7. Let A be a separable C∗ -algebra, (X, µ) a Lebesgue space, X x → ϕx and X x → ψx measurable maps into the state space of A, the state space being considered with the weak∗ topology. Consider the states ϕ and ψ on A ⊗ L∞ (X, µ) deﬁned by ϕ(a ⊗ f ) = ϕx (a)f (x)dµ(x), ψ(a ⊗ f ) = X ψx (a)f (x)dµ(x).