Dynamical Entropy in Operator Algebras. Ergebnisse der by Sergey Neshveyev

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By Sergey Neshveyev

The booklet addresses mathematicians and physicists, together with graduate scholars, who're attracted to quantum dynamical structures and functions of operator algebras and ergodic conception. it's the purely monograph in this subject. even if the authors imagine a easy wisdom of operator algebras, they provide targeted definitions of the notions and quite often entire proofs of the consequences that are used.

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Extra info for Dynamical Entropy in Operator Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge

Example text

In−1 with Ik = Ik for k < n − 1 and In−1 = In−1 ×In . In other words, any abelian model (C, µ, {Ck }nk=1 , P ) for n−1 (A, ϕ, {γk }nk=1 ) defines an abelian model (C, µ, {Ck }n−1 k=1 , P ) for (A, ϕ, {γk }k=1 ) with Ck = Ck for k < n − 1 and Cn−1 = Cn−1 ∨ Cn . Comparing the entropies of these abelian models, we see that the classical terms as well as the first n−2 correction terms coincide. 8 applied to I = In−1 and J = In shows that the entropy of (C, µ, {Ck }n−1 k=1 , P ) is not smaller than the entropy of (C, µ, {Ck }nk=1 , P ).

N ). So far we have not given any estimates for Hϕ . 4. For any channel γ: B → A, we have 0 ≤ Hϕ (γ) ≤ S(ϕ ◦ γ). If, moreover, A is a von Neumann algebra, ϕ a faithful normal state and the image of γ does not consist only of scalars, then Hϕ (γ) > 0. 3(iii) this implies that 0 ≤ Hϕ (γ1 , . . 3(i),(iv) Hϕ (γ1 , . . , γn ) ≤ Hϕ (B, . . , B ) = Hϕ (B) ≤ S(ϕ|B ). 4. 2)) ϕi (1)S(ϕˆi ◦ γ, ϕ ◦ γ) = S(ϕ ◦ γ) − i ϕi (1)S(ϕˆi ◦ γ). i The left hand side of the above equality shows that Hϕ (γ) ≥ 0, while the right hand side implies Hϕ (γ) ≤ S(ϕ ◦ γ).

N and γ1 , . . in we have to estimate the differences between the correction terms. Thus, what we need to prove is the following: there exists δ > 0 such that if γ, γ : B → A are channels, dim B ≤ d, γ − γ ϕ < δ and ϕ = i∈I ϕi with |I| ≤ r, then S(ϕi ◦ γ, ϕ ◦ γ) − i S(ϕi ◦ γ , ϕ ◦ γ ) < i We have S(ϕi ◦ γ, ϕ ◦ γ) − i S(ϕi ◦ γ , ϕ ◦ γ ) i ε . 1 Mutual Entropy = S(ϕ ◦ γ) − S(ϕ ◦ γ ) − 47 ϕi (1)(S(ϕˆi ◦ γ) − S(ϕˆi ◦ γ )). i Choose δ0 > 0 such that |S(ψ1 ) − S(ψ2 )| < ε/6 for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ0 and dim B ≤ d.

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