Calculus on manifolds by Spivak, M
By Spivak, M
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Extra info for Calculus on manifolds
The ﬁrst equality in (4) is clear from (1), and the rest is clear from the string ¯ r ⊗ X ⊗ K c )∗ ∼ (H = CB(X, CB(K c , H c )), = CB(X ⊗ K c , H c ) ∼ which equals CB(X, B(K, H)). 14). 14). Then 38 Duality and tensor products ¯ r by commutativity of ⊗min , and so S ∞ (K, H) ⊗min X = H c ⊗min X ⊗min K the second part of (5) follows from (2). Item (6) is a special case of (5), and (7) follows from (6) by (3) and the associativity of the Haagerup tensor product. The middle equality in (8) follows from (2), and the ﬁrst equality from (1).
9. Suppose that c ∈ A, and that c satisﬁes u(c)∗ u(c) = u(c∗ c). Then u(ac) = u(a)u(c) for all a ∈ A. Proof Suppose that B ⊂ B(H). We write u = V ∗ π(·)V as in Stinespring’s theorem, with V ∗ V = IH . Let P = V V ∗ be the projection onto V (H). By hypothesis V ∗ π(c)∗ P π(c)V = V ∗ π(c)∗ π(c)V . For ζ ∈ H, set η = π(c)V ζ. Then P η 2 = V ∗ π(c)∗ P π(c)V ζ, ζ = η 2 . Thus P η = η, and V V ∗ π(c)V = π(c)V . Therefore u(a)u(c) = V ∗ π(a)V V ∗ π(c)V = V ∗ π(a)π(c)V = u(ac). 9, and that there is a C ∗ -subalgebra C of A with 1A ∈ C, such that π = u|C is a ∗-homomorphism.
Let u˜ : X ⊗ Y → W be the canonically associated linear map. 40). 6 and the fact at the end of the second last paragraph, we see that the bilinear map (x, y) → ϕ(x)ψ(y) is completely contractive. 1), we deduce that the latter norm of an element z ∈ X ⊗ Y is dominated by z h. Hence indeed · h is a norm. By the fact at the end of the last paragraph, together with Ruan’s theorem, we see that the completion X ⊗h Y of X ⊗ Y with respect to · h is an operator space. This operator space is called the Haagerup tensor product.