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

By Pere Ara, Fernando Lledo, Francesc Perera

This quantity includes survey papers at the concept of operator algebras in keeping with lectures given on the ""Lluis Santalo"" summer season tuition of the genuine Sociedad Matematica Espanola, held in July 2008 on the Universidad Internacional Menendez Pelayo, in Santander (Spain).

Topics during this quantity hide present basic facets of the speculation of operator algebras, that have very important functions such as:

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

The concept of operator algebras, brought within the thirties by means of J. von Neumann and F. J. Murray, was once built in shut dating with primary elements of useful research, ergodic concept, harmonic research, and quantum physics. extra lately, this box has proven many different fruitful interrelations with numerous parts 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

N n Proof. Write xn = [Xn ]. 15, modules Xn,m such that Xn,1 ⊂⊂ Xn,2 ⊂⊂ · · · ⊂⊂ Xn , and Xn = ∪m Xn,m . Since we have [Xn,m ] ⊂⊂ [Xn ] ≤ [Xn+1 ], we obtain that [Xn,m ] ⊂⊂ [Xn+k ] for all k ≥ 0. Thus [Xn,m ] ∈ x⊂⊂ n+k . 16 to find l = l(n, m) such that [Xn,m ] ⊂⊂ [Xn+1,l ]. By using this observation, we will construct the sequence (xn ). Find l1 such that [X1,1 ] ⊂⊂ [X2,l1 ]. Reindexing the sequence [X2,n ] (and throwing out a finite number of the X2,∗ ’s), we may assume that l1 = 2. Next, there is l2 such that [X2,2 ] , [X1,2 ] ⊂⊂ [X3,l2 ] .

I=N This shows that the inner product is well-defined. Now we will check the (n) completeness of Xi . Let x(n) = (xi ) ∈ Xi be a Cauchy sequence. Given > 0 there is N0 such that for n, m ≥ N0 we have x(n) − x(m) 2 < . Thus for n, m ≥ N0 we get ∞ (2) (n) xi (m) − xi (n) , xi (m) − xi < . i=1 (n) (m) (n) (m) < for n, m ≥ N0 , and In particular, given i, we have xi − xi , xi − xi (n) we see that there exists xi = limn xi ∈ Xi . Set x = (x1 , x2 , . . ). We want to see that x ∈ i Xi and that x = limn x(n) .

Let X be a nonempty separable compact Hausdorff space, and choose elements x1 , x2 , . . ∈ X such that {xn , xn+1 , . } is dense in X for each n. For all positive integers n and k, let δn : Mk (C(X)) → Mk (C) ⊆ Mk (C(X)) be the C∗ -homomorphism given by evaluation at xn . Let υ(1), υ(2), . . be positive integers such that υ(n) | υ(n + 1) for all n, and set An = Mυ(n) (C(X)) for each n. We choose maps φn : An → An+1 of the following form: φn (a) = (a, . . , a, δn (a), . . , δn (a)) . Let αn the number of identity maps involved in the definition of φn , and set φs,n = φs−1 φs−2 .

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