By David P. Blecher
This worthy reference is the 1st to provide the final thought 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 fundamental strategies, concept and methodologies had to equip a starting researcher during this sector. a tremendous pattern in smooth arithmetic, encouraged mostly by means of physics, is towards `noncommutative' or `quantized' phenomena. In sensible research, this has seemed particularly below the identify of `operator spaces', that is a version of Banach areas that's really applicable for fixing difficulties relating areas or algebras of operators on Hilbert house bobbing up in 'noncommutative mathematics'. the class of operator areas contains operator algebras, selfadjoint (that is, C*-algebras) or another way. additionally, many 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 area conception, is the most subject of the e-book. A basic conception of operator algebras, and their modules, certainly develops out of the operator house technique. certainly, operator area concept is a delicate sufficient medium to mirror properly many very important non-commutative phenomena. utilizing fresh advances within the box, the booklet indicates how the underlying operator house constitution captures, very accurately, the profound relatives among the algebraic and the useful analytic buildings concerned. the wealthy interaction among spectral conception, operator thought, C*-algebra and von Neumann algebra recommendations, and the inflow of vital principles from similar disciplines, equivalent to natural algebra, Banach house conception, Banach algebras, and summary functionality concept is highlighted. every one bankruptcy ends with a long component of notes containing a wealth of extra information.
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Additional resources for Operator Algebras and Their Modules: An Operator Space Approach
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35), we regard X ⊗min Y → CB(Y∗ , X), and we define the normal minimal tensor product X ⊗ Y to be the w ∗ -closure of X ⊗Y (or of X ⊗min Y ) in CB(Y∗ , X). Equivalently, if X and Y are w∗ -closed subspaces of B(H) and B(K) respectively, then we may define X ⊗ Y to be the w ∗ -closure in B(H ⊗2 K) of the copy of X ⊗ Y . This is sometimes referred to as the normal spatial tensor product. g. see [407, IV; 5]) 40 Duality and tensor products and in particular, B(H) ⊗ B(K) = B(H ⊗2 K). To see that these two definitions of X ⊗ Y are the same (up to w∗ -homeomorphic complete isometry), we use the following argument.
Using the separability assumption it is easy to see that for such an f , the scalarvalued function f (· ) Y is measurable. We set f ∞ = f (· ) Y L∞ (Ω) and we say that f is essentially bounded if f ∞ < ∞. Then f ∞ is a seminorm on the space of such functions and we let L∞ (Ω; Y ) be the normed space obtained after Operator spaces 41 taking the quotient by N0 = {f : f ∞ = 0}. As in the scalar-valued case, we identify any w∗ -measurable essentially bounded function f with its class modulo N0 . Using again the separability of Y∗ , we see that for any f ∈ L∞ (Ω; Y ), f ∞ = sup f (· ), ϕ L∞ : ϕ ∈ Ball(Y∗ ) .
11). Thus the second last centered equation exactly means that the map un from Mn (Min(G)) to Mn (H r ) has norm ≤ π2 (u). Thus u cb ≤ π2 (u). The converse inequality is easily obtained by reversing the arguments. 30). 45) with G = E ∗ we see that if w : E ∗ → 2m is the linear mapping corresponding to x, then x is equal to π2 (w). 6), show that Max(E) ⊗h Min(F ) = E ⊗g2 F isometrically. 46) isometrically. 11 (Operator space projective tensor product) As with the Haagerup tensor product, it is convenient to first define an intimately related class of bilinear maps.