The Classification Of Extensions Of Certain C～*-algebras

The theory of extension of C~*-algebras came from operator theory. If H denotes the separable, infinite-dimensional Hilbert space, and K the closed ideal of compact operators in B(H), then we have a short exact sequence 0→ K→ B(H) →|πQ→0, where Q denotes the Calkin algebra B(H)/K,π denotes the natural map.The problem in operator theory center around the fact that some elements x in B(H) are not invertible, even though their images π(T) in Q have this property. In other word, Fredholm operator in B(H) is not always invertible. The Fredholm index (index(T) = dimkeriT) — dimker(T~*))turns to be the only obstruction to invertibility.Let A and B be C~*-algebras, an extension of A by B is a short exact sequence 0→ B→ E→ A→0 of C~*-algebras. Given A and B, the goal of extension theory is to classify all extensions of A by B up to a suitable notion of equivalence. The abstract theory of extensions began in 1968 with Busby's thesis. Busby found that any extension of A by B can be recovered from its Busbyinvariant(the homomorphism r from A to Q(B)), then we often identify an extension with its Busby invariant. The strong isomorphism equivalence used by Busby seems to be rarely useful in practice, then there were another equivalence, which include strong(unitary)equivalence, weak(unitary)equivalence and homotopy equivalence. For every equivalence, there were some excellent conclusions about C*-algebra extensions.Breuer first associated an index map to extensions other than those of the compact operators. The boundary map Si from the six term exact sequence associated with extension takes the role of index map. Later Brown used the boundary 50 in the problem of lifting projections. From then on, The extension theory was associated with K-theory completely.During the past decade, the classification of C*-algebras have rapid development. These algebras include AH algebra and the algebras with tracial topological rank. In the classification of C*-algebras, we often require the C*-algebra to be closed under extension. Then we have the following questions:Question one: For AH algebras and the C*-algebras with tracial topological rank, which conditions can assure that they are closed under extension?L.G.Brown proved that AF algebra is closed under extension. But it is not true for AT algebra. Professor Lin proved that let A and B be AT algebras of real rank zero, E be an extension of B by A, when So,5i are both zero, E is an AT algebras of real rank zero. In the second chapter, we considerthe ordinary AT algebra and AI algebra without real rank zero, and get the following conclusions.Theorem 2.2.1: Let A be an AT algebra with real rank zero, B be an AT algebra; then the extension E of B by A is an ^4T algebra if and only if the extension is quasidiagonal.Theorem 2.3.1: Let A be an AI algebra which has an approximate unit of projections, B be an AI algebra; then the extension E of B by A is an AI algebra if and only if the extension is quasidiagonal.Theorem 2.3.2: Let A be an AT algebra which has an approximate unit of projections, B be an AT algebra; then the extension E of B by A is an AT algebra if and only if the extension is quasidiagonal.For TAF algebra, Professor Lin proved that it is closed under quasidiagonal extension. In the third chapter, we consider the C*-algebra with tracial topological rank no more than 1, and get the following conclusion.Theorem 3.2.1: Let J and B be separable simple C*-algebras with TR{A) < 1,TR(B) < 1. If the extension is unital quasidiagonal, then TR{A) < 1.Because the invariants of AH algebra and TAF algebra are K* group, and the extension theory are associated with if-theory completely, then we have the following question.Question two: Whether the K* group can be used to classify the C*-algebra extension or not.In the second chapter, we consider an especial AH algebra extension. Using K-theory and KK-theory, we get the following conclusion.Proposition 2.4.3: Let 0 -f K -f E{ A B{ ->? 0(* = 1,2) be unital and essential quasidiagonal extension, K, be the compact operators and Bi be one dimensional simple AH algebras with RR(Bi) = 0, Suppose that there exists an order isomorphism 9 : (K*(Ei), D*(Ey)) —> (K+(E2),D*(E2)), then there exists a ^-isomorphism a : E± —?? E2 such that a+ = 9.In the third chapter, we consider an especial extension of TAF algebra. Just like the proof in the second chapter, we get the following conclusion.Theorem 3.3.2: Let 0 -?? K -?? Et ^ Bi -?? 0(i = 1,2) be unital essential quasidiagonal extension, K, be the compact operators and B{ be separable, simple and nuclear C*-algebra such that TR(Bi) = 0. Suppose that UCT is true for Bi,Bi (K*(E2),D*(E2))i there exists a ^-isomorphism a : Ei —> ￡"2 such that a* = ^.