Extensions By C*-Algebras With Continuous Scale

In 1990s,H.Lin gave the definition of continuous scale C*-algebra,and classi-fied the extensions of certain commutative C*-algebras by C*-algebras with continuous scale.Since a C*-algebra with continuous scale is not stable,the classification of C*-algebra extensions by such C*-algebras can be viewed as a generalization of the classical”BDF Theory”to the non-stable case.In this dissertation,we work on the extensions of UHF-algebra of 2^∞-type and Jiang-Su algebra by C*-algebras with continuous scale,and classify such extensions up to unitary equivalence.In Chapter 1,we introduce the background of C*-algebra extension theory,and the progress of research on the classification of extensions by C*-algebras with continuous scale.In Chapter 2,we mainly introduce some properties related to C*-algebras with continuous scale,and the characterization of quasidiagonal extensions,also some basic lemmas.In Chapter 3,we work on the extentions of UHF algebra A of 2^∞-type by B（？）A,where B is a non-unital butσ-unital,simple C*-algebra with real rank zero,stable rank one,K₀（B）weakly uperforated and a continuous scale.In order to characterize the zero element of the abelian group Ext（A,B（？）A）,we construct a special kind of trivial extensions,denoted byη_p.And we classify these trivial extensions up to unitary equiv-alence.We also prove that in the setting of this chapter,all the quasidiagonal extensions are trivial.Since trivial extensions are quasidiagonal,then triviality is equivalent qua-sidiagonality.In the end of this chapter,we give an isomorphism of the abelian groups Ext（A,B（？）A）and KK（A,M（B（？）A）/B（？）A）.In Chapter 4,we work on the extensions of Jiang-Su algebra,in which the ideal is non-unital butσ-unital,simple,nuclear C*-algebra with tracial rank zero,Z-stability and a continuous scale.We also construct a kind of trivial extensionsη_p,and then,with the same method that used in Chapter 3,we classify the zero element in Ext（Z,B）.We give a uniqueness theorem of maps from ASH algebra to C*-algebras with tracial rank zero,which is used in the proof of Theorem 4.3.1.This uniquess theorem is a simplification of a more general one in[32].In the end,we conlude that there is an isomorphism of abelian groups from Ext（Z,B）to KK（Z,M（B）/B）).