Aspects of Operator Algebras and Applications: Uimp-rsme by Pere Ara, Fernando Lledo, Francesc Perera

By Pere Ara, Fernando Lledo, Francesc Perera

This quantity comprises survey papers at the concept of operator algebras according to lectures given on the ""Lluis Santalo"" summer season institution of the true Sociedad Matematica Espanola, held in July 2008 on the Universidad Internacional Menendez Pelayo, in Santander (Spain).

Topics during this quantity conceal present basic features of the speculation of operator algebras, that have very important purposes such as:

* $K$-Theory, the Cuntz semigroup, and class for $C^*$-algebras * Modular conception for von Neumann algebras and functions to Quantum box concept * Amenability, Hyperbolic teams, and Operator Algebras.

The conception of operator algebras, brought within the thirties via J. von Neumann and F. J. Murray, was once constructed in shut courting with basic elements of practical research, ergodic conception, harmonic research, and quantum physics. extra lately, this box has proven many different fruitful interrelations with a number of components of arithmetic and mathematical physics.

This publication is released in cooperation with genuine Sociedad Matematica Espanola (RSME).

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Extra info for Aspects of Operator Algebras and Applications: Uimp-rsme Lluis a Santalo Summer School, Universidad Internacional Menendez Pelayo, Santander, Spain, July 21-25, 2008

Example text

Ai , and let k ≥ max{n, m}. Then: Let K = sup 1≤i≤m m ϕk+1 (xi ) − ϕk (xi ) K ≤ Km2−k , (n) ϕk+1 (x) − ϕk (x) ≤ (n) i=1 which implies that the sequence (ϕk (x)) is Cauchy. Thus ϕ defines a map on S which is isometric (we just checked it is well defined and is isometric as each ϕk is), and so it extends to a Hilbert module map ϕ : X → Y , isometric onto its image. (ii). Assume now that [X] = [Y ]. Working as in (i), now with the two inequalities [X] ≤ [Y ] and [Y ] ≤ [X], and except maybe passing to subsequences of the Xi ’s and the Yj ’s, we obtain maps ϕn : Xn → Yn and ψn : Yn → Xn+1 (isometries onto their images) We shall modify as in (i) the given maps.

Then (i) [X] ≤ [Y ] in Cu(A) if and only if X ∼ =X ⊆Y. (ii) [X] = [Y ] if and only if X ∼ =Y. Proof. Write X = ∪∞ i=1 Xi , where the Xi ’s are countably generated, X1 ⊂⊂ X2 ⊂⊂ · · · ⊂⊂ X , and likewise Y = ∪∞ i=1 Yi for countably generated modules Yi with Y1 ⊂⊂ Y2 ⊂⊂ · · · ⊂⊂ Y . 12 that [X] = sup[Xi ] and [Y ] = sup[Yi ]. i i (i). We only need to show that, if [X] ≤ [Y ], then X ∼ =X ⊆Y. [X] ≤ [Y ] = sup[Yi ]. Therefore, there is an Since X2 ⊂⊂ X, we have [X2 ] i index i such that [X2 ] ≤ [Yi ], and since X1 ⊂⊂ X2 , we have by the order-relation in Cu(A) that there is an (isometric) isomorphism of X1 onto a compactly contained submodule of Yi .

Hilbert C*-modules. 1. Introduction. This is an overview of some standard facts on Hilbert C*modules. Good references for this section are the books by Manuilov and Troitsky [45] and Lance [42]. In particular we have followed (parts of) Chapters 1 and 2 of the book [45] in our exposition. 2. Hilbert C∗ -modules. Hilbert modules were introduced in the ”Appendix” of this book [1]. 1. Let A be a C∗ -algebra. A (right) Hilbert A-module is a right A-module X together with an A-valued inner product X × X → A, (x, y) → x, y , enjoying the properties (i) (ii) (iii) (iv) x, αy + βz = α x, y + β x, z , x, ya = x, y a, ∗ y, x = x, y x, x ≥ 0 and x, x = 0 if and only if x = 0.

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