Operator Algebras and Their Modules: An Operator Space by David P. Blecher

By David P. Blecher

This beneficial reference is the 1st to give the final idea of algebras of operators on a Hilbert house, and the modules over such algebras. the recent conception of operator areas is gifted early on and the textual content assembles the elemental thoughts, concept and methodologies had to equip a starting researcher during this sector. an immense development in sleek arithmetic, encouraged mostly by way of physics, is towards `noncommutative' or `quantized' phenomena. In useful research, this has seemed significantly below the identify of `operator spaces', that is a version of Banach areas that is fairly applicable for fixing difficulties pertaining to areas or algebras of operators on Hilbert house bobbing up in 'noncommutative mathematics'. the class of operator areas comprises operator algebras, selfadjoint (that is, C*-algebras) or in a different way. additionally, lots of the vital modules over operator algebras are operator areas. a typical therapy of the themes of C*-algebras, Non-selfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), jointly less than the umbrella of operator house idea, is the most subject of the publication. A common thought of operator algebras, and their modules, certainly develops out of the operator area technique. certainly, operator area conception is a delicate sufficient medium to mirror competently many vital non-commutative phenomena. utilizing fresh advances within the box, the publication indicates how the underlying operator house constitution captures, very accurately, the profound relatives among the algebraic and the sensible analytic constructions concerned. the wealthy interaction among spectral thought, operator idea, C*-algebra and von Neumann algebra options, and the inflow of vital rules from comparable disciplines, comparable to natural algebra, Banach house idea, Banach algebras, and summary functionality concept is highlighted. each one bankruptcy ends with a long portion of notes containing a wealth of extra information.

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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.

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