# 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.

Read Online or Download Dynamical Entropy in Operator Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge PDF

Best linear books

Constrained Optimal Control of Linear and Hybrid Systems

Many sensible keep watch over difficulties are ruled by way of features resembling kingdom, enter and operational constraints, alternations among diversified working regimes, and the interplay of continuous-time and discrete occasion platforms. at this time no method is accessible to layout controllers in a scientific demeanour for such structures.

The semicircle law, free random variables and entropy

The booklet treats unfastened chance thought, which has been greatly built because the early Eighties. The emphasis is wear entropy and the random matrix version process. the amount is a special presentation demonstrating the large interrelation among the subjects. Wigner's theorem and its vast generalizations, equivalent to asymptotic freeness of self sufficient matrices, are defined intimately.

Limit Algebras: An Introduction to Subalgebras(Pitman Research Notes in Mathematics Series, 278)

Written through one of many key researchers during this box, this quantity develops the idea of non-self adjoint restrict algebras from scratch.

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

Sample text

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.