Cbms Lecture Series
نویسنده
چکیده
Notes taken by Stuart White during the lectures and not proof read. Errors will abound. 1. Nuclearity: Approximation and perturbation Recall: Definition 1.1. A C∗-algebra. Definition 1.2. A cp map Theorem 1.3. Stinespring’s theorem. Every cp map is a compression of a ∗-homomorphism. Definition 1.4. φ : A → B cp has order zero if a ⊥ b =⇒ φ(a) ⊥ φ(b). 1 Theorem 1.5. (i) φ : A→ B has order zero iff φ(·) = hπ(·) = π(·)h, where π : A→M(C∗(φ(A))) ⊆ B∗∗ is a ∗-hm and h ∈M(C(φ(A)))+∩ φ(A)′. When A is unital h = φ(1A). (ii) CPCord0(A,B) ↔ Hom(C0((0, 1], A), B)) is a one-one correspondence. (iii) φ : A → B be cp order zero, f ∈ C0(0, ‖φ‖], then we can define a new cp order zero map f(φ) : A → B by f(φ)(a) = f(h)π(a). This gives a functional calculus to order zero maps. Order zero maps are precisely those compressions of ∗-hms by positive elements which commute with the range. A unital order zero map is just a ∗-homomorphism. The bijection in (ii) arises as C0((0, 1] is the universal C∗-algebra generated by a positive contraction (this corresponds to the h). Recall: Date: May 15, 2012. 1Positive elements a, b are orthogonal if ab = 0. 1
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