By Oertel, Herbert

Dieses Buch wendet sich an Studenten der Ingenieurwissenschaften und Ingenieure der Raumfahrtindustrie und der Energieverfahrenstechnik. Es verkn?pft die klassischen Gebiete der Aerodynamik mit der Nichtgleichgewichts-Thermodynamik hei?er Gase. Am Beispiel des Wiedereintritts einer Raumkapsel in die Erdatmosph?re werden die aerothermodynamischen Grundlagen und numerischen Methoden zur Berechnung des Str?mungsfeldes der Raumkapsel im gaskinetischen und kontinuumsmechanischen Bereich der Wiedereintrittstrajektorie behandelt. Am Beispiel von Raumfahrtprojekten werden die Methoden entwickelt. Die Autoren sind anerkannte Spezialisten f?r dieses Fachgebiet.

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1(vi), S(ϕ ◦ γi ◦ θi , ψ ◦ γi ◦ θi ) ≤ S(ϕ ◦ γi , ψ ◦ γi ) ≤ S(ϕ, ψ). Hence S(ϕ ◦ γi , ψ ◦ γi ) → S(ϕ, ψ). 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.

In }) the mutual entropy of γ1 , . . in . 2) ˆ where for a positive linear functional ψ we denote by ψ the state ψ(1)−1 ψ. ,in (k) Hϕ (γk ; {ϕik }). 1 Mutual Entropy 35 In the case when the Ai ’s are subalgebras of A and the γi ’s are the inclusion maps, we write Hϕ (A1 , . . , An ) instead of Hϕ (γ1 , . . , γn ). To get a better understanding of the definition, consider the case of one channel γ: B → A. Then Hϕ (γ) is the supremum of the quantities S(ϕi ◦ γ, ϕ ◦ γ). 4) i For a state ψ on B, call a decomposition ψ = i ψi orthogonal if the density matrices of the ψi ’s are mutually orthogonal.

In particular, the fixed point algebra M S∞ is trivial. On the other hand, since any automorphism of a full matrix algebra is inner, for each g ∈ S∞ there exists ug ∈ ∪n B[−n,n] such that αg = Ad ug . Since ϕ is αg -invariant, we have ug ∈ Mϕ = N . Hence if z is in the center of N , then z ∈ M S∞ . Therefore z is a scalar. For the second proof note that M , being an infinite tensor product of factors with respect to a product-state, is a factor. Thus it suffices to check that the center of N is contained in the center of M .

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