An introduction to the classification of amenable by Huaxin Lin

By Huaxin Lin

The idea and functions of C*-algebras are regarding fields starting from operator idea, team representations and quantum mechanics, to non-commutative geometry and dynamical structures. by way of Gelfand transformation, the idea of C*-algebras is additionally 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) through their K-theoretical facts. It begun with the category of AT-algebras with actual rank 0. because then nice efforts were made to categorise amenable C*-algebras, a category of C*-algebras that arises so much obviously. for instance, a wide type of easy amenable C*-algebras is came across to be classifiable. the applying of those effects to dynamical platforms has been confirmed.

This e-book 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 speculation of C*-algebras that are relatively vital to the idea of the category of amenable C*-algebras. bankruptcy four otters the category of the so-called AT-algebras of genuine rank 0. the 1st 4 chapters are self-contained, and will function a textual content for a graduate direction on C*-algebras. The final chapters include extra complex fabric. specifically, they care for the class theorem for easy AH-algebras with genuine rank 0, the paintings of Elliott and Gong. The booklet includes many new proofs and a few unique effects relating to the type of amenable C*-algebras. in addition to being as an creation to the idea of the class of amenable C*-algebras, it's a finished reference for these extra accustomed to the topic.

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Moreover, B/{Br\I)^(B + I)/I. Proof. Clearly B + I is a *-subalgebra of A containing / and B. Let 7T : A —> A/I be the quotient map. , B + I is closed. Note that the map 6 + JBn/h->6 + / i s a *-isomorphism from B/(Bf)I) to (B + I)/I. 15, it is a C*-isomorphism. 1 A linear map 4> : A —»• B between C*-algebras is said to be self-adjoint if (Asa) C Bsa, and positive if (A+) C B+. It follows that if <> / is positive then is self-adjoint. Every homomorphism h : A —> B is positive. If B = C, then a positive linear map : A -4 C is called a positive linear functional.

So the weak operator topology for A" is sometimes called cr-weak topology. 1 Let {TT\,HI) and (7r 2 ,iJ 2 ) be two cyclic representations of a C*-algebra A with cyclic vectors £1 and £2 (||£i|| = H&H)- Then there exists an isometry u : H\ —>• H2 such that m = U*TT2U with u(£i) = £2 if and only if {ni(a)£i,£i) = (7r2(a)£2,£2) for all a £ A. Proof. If u£i = & then (i"i(a)£i,6) = (u*7T 2 (aXi,fi) = (7r 2 (a)6,6> for all a G A. Conversely, define a linear map u from u(ni(a)£i) = 7r2(a)£2. Since TTI(A)£I onto TT 2 (A)£ 2 ll«M«)fc)l| 2 = (7r 2 (o*a)6,6> = M a ' a ) £ i , 6 ) = IKi(aKil| 2 , by Enveloping von Neumann algebras and the spectral theorem 39 we see that u extends to an isometry from the closure of iti(A)£i onto the closure of ^(A)^Since £j and £2 are cyclic vectors, u is an isometry from H\ onto Hi.

Put b = f ° * ) . Then 6 G Af 2 (yl") so . Therefore, from what we have shown, ||J 2 (6)|| = IHI = ||z||. So \J{x){f)\ = \J2{b){ct>)\<\\b\\U\\ = \\x\\\\f\\. This implies that ||J(a;)|| < ||ir||. Therefore J is an isometry. Since A is weakly dense in A", J{A) is dense in J(A") in the weak*-topology as a subset of (A*)*. However, j(A) is dense in (A*)* in the weak*-topology. Therefore, since J (a) = 3 (a) for a G A, J {A") = (A*)* = A**. D The following corollary follows from the above theorem and the uniform boundedness theorem.

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