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By Huaxin Lin

The idea and purposes of C*-algebras are with regards to fields starting from operator concept, team representations and quantum mechanics, to non-commutative geometry and dynamical platforms. via Gelfand transformation, the idea of C*-algebras can be considered as non-commutative topology. a few decade in the past, George A. Elliott initiated this system of class of C*-algebras (up to isomorphism) by means of their K-theoretical facts. It all started with the type of AT-algebras with actual rank 0. considering the fact that then nice efforts were made to categorise amenable C*-algebras, a category of C*-algebras that arises such a lot certainly. for instance, a wide type of straightforward amenable C*-algebras is came upon to be classifiable. the applying of those effects to dynamical platforms has been tested.

This ebook introduces the new improvement of the speculation of the class of amenable C*-algebras ? the 1st such try. the 1st 3 chapters current the fundamentals of the idea of C*-algebras that are relatively vital to the idea of the type of amenable C*-algebras. bankruptcy four otters the category of the so-called AT-algebras of actual rank 0. the 1st 4 chapters are self-contained, and will function a textual content for a graduate path on C*-algebras. The final chapters include extra complex fabric. particularly, they take care of the category theorem for easy AH-algebras with actual rank 0, the paintings of Elliott and Gong. The ebook comprises many new proofs and a few unique effects concerning the class of amenable C*-algebras. in addition to being as an advent to the idea of the class of amenable C*-algebras, it's a accomplished reference for these extra acquainted with the topic.

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Then there is a unique spectral measure E relative to (sp(a),H) such that a= f XdEx. Jsp(o) Proof. By the Borel functional calculus, the embedding j : C*(l,a) —> B{H) extends to a weak-weak continuous homomorphism j : B(X) —• B(H). For each Borel subset of sp(a), j(xs) = E(S) is a projection. It is easy to verify that {E(S) : S Borel} forms a spectral measure. Since the identity function on sp(a) is Borel, it follows from the Borel functional Enveloping von Neumann algebras and the spectral theorem 41 calculus that a= f XdEX- •/sp(a) We leave it to the reader to check the uniqueness of E.

5 Every C*-algebra A admits an approximate identity. Indeed, if A is the upwards-directed set of all a G A+ with \\a\\ < 1 and e\ = A for all A G A, £/ien {e\)\e^ forms an approximate identity for A. Proof. 3 {e\} is an increasing net in the closed unit ball of A. We need to show that lim^ ae\ = a for all a £ A. Since A spans A, it suffices to assume that a G A+. Since {||a(l — eA)a||} is decreasing, it suffices to show that there are un G {e\} such that ||a(l — u„)a|| —¥ 0 (as n -> oo). Note that un = (1 - ^)fi(a) G A.

Set £ = & © • • • © £„ € HW. It is easily checked that p(M)' = {a G B(H^) : ay G M ' } . Therefore p(x) G p(M)". From what we have proved in the first part of the proof, we obtain a G M such that n £ | | ( z - a ) a | | 2 = |l(p(z)-Ka)KI| 2 <£ 2 . fc=i It follows that x is in the strong closure of M. Thus x G M. In other words, M " C M. Since clearly M cM",M = M". 9 A weakly closed C*-subalgebra M C B(H) is called a von Neumann algebra. In other words, a C*-subalgebra M C B{H) is a von Neumann algebra if M = M" and 1 G M.

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