Dynamical Entropy in Operator Algebras. Ergebnisse der by Sergey Neshveyev

By Sergey Neshveyev

The ebook 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 merely monograph in this subject. even though the authors think a easy wisdom of operator algebras, they offer distinct definitions of the notions and regularly whole proofs of the implications that are used.

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Additional resources for Dynamical Entropy in Operator Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge

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2 Entropy of Dynamical Systems 53 It remains to prove the claims. If y ∈ ∪k B[−k,k] , then ϕ(αn (xx∗ )y) = ϕ(xx∗ )ϕ(y) if n is sufficiently large. Thus any weak operator limit point a of the sequence {αn (xx∗ )}n has the property ϕ(ay) = ϕ(xx∗ )ϕ(y). Hence a = ϕ(xx∗ )1, and the first claim is proved. To prove the second claim, note 1/2 that αn (x)ξϕ = Jϕ ∆ϕ αn (x∗ )ξϕ = λ−1/2 Jϕ αn (x∗ )ξϕ , where we use the same notation as on page 35.

A related but easier result is the following. 6. If {Ai }i is an increasing net of unital C∗ -subalgebras of a C∗ -algebra A such that ∪i Ai is norm-dense, then S(ϕ|Ai , ψ|Ai ) S(ϕ, ψ) for any ϕ and ψ. 28 2 Relative Entropy Proof. 10), since it suffices to consider step functions with values in ∪i Ai . Recall that a unital C∗ -algebra is called nuclear if there exist two nets {γi : Ai → A}i and {θi : A → Ai }i of unital completely positive maps with finite dimensional C∗ -algebras Ai such that (γi ◦ θi )(a) − a → 0 for every a ∈ A.

7) follows from (v) applied to the state ϕ. ˜ To prove (vii) note that as B is maximal abelian in A, we have TrA |B = TrB . In particular, if EB : A → B is the trace preserving conditional expectation, then Qϕ◦EB = Qϕ|B . Hence S(ϕ|B ) − S(ϕ) = ϕ(log Qϕ − log Qϕ|B ) = S(ϕ, ϕ ◦ EB ). 2(vi). Hence S(ϕ|B ) ≥ S(ϕ). If Qϕ ∈ B, then Qϕ|B = Qϕ as TrA |B = TrB , so S(ϕ|B ) = S(ϕ). Conversely, if S(ϕ|B ) = S(ϕ), then S(ϕ, ϕ◦EB ) = 0. 2(i) we get ϕ = ϕ◦EB , so Qϕ = Qϕ◦EB = Qϕ|B ∈ B, that is, B is in the centralizer of ϕ.

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