# 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.

**Read or Download Operator Algebras and Their Modules: An Operator Space Approach PDF**

**Best linear books**

**A first course in linear algebra**

A primary path in Linear Algebra is an advent to the elemental innovations of linear algebra, in addition to an creation to the thoughts of formal arithmetic. It starts with structures of equations and matrix algebra prior to stepping into the speculation of summary vector areas, eigenvalues, linear changes and matrix representations.

**Measure theory/ 3, Measure algebras**

Fremlin D. H. degree thought, vol. three (2002)(ISBN 0953812936)(672s)-o

**Elliptic Partial Differential Equations**

Elliptic partial differential equations is among the major and such a lot energetic parts in arithmetic. In our publication we research linear and nonlinear elliptic difficulties in divergence shape, with the purpose of supplying classical effects, in addition to newer advancements approximately distributional ideas. accordingly the ebook is addressed to master's scholars, PhD scholars and somebody who desires to commence study during this mathematical box.

- Linear Dynamical Systems
- Introduction to Quantum Groups and Crystal Bases (Graduate Studies in Mathematics)
- Nonlinear Predictive Control: Theory and Practice (Iee Control Series, 61)
- Riesz and Fredholm Theory in Banach Algebras (Chapman & Hall/CRC Research Notes in Mathematics Series)

**Additional resources for Operator Algebras and Their Modules: An Operator Space Approach**

**Sample text**

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.