C-star-algebras by Jacques Dixmier

By Jacques Dixmier

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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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