C*-Algebra Extensions and K-Homology by Ronald G. Douglas

By Ronald G. Douglas

Recent advancements in diversified components of arithmetic recommend the research of a definite classification of extensions of C*-algebras. right here, Ronald Douglas makes use of tools from homological algebra to check this selection of extensions. He first exhibits that equivalence periods of the extensions of the compact metrizable house X shape an abelian team Ext (X). moment, he exhibits that the correspondence X ? Ext (X) defines a homotopy invariant covariant functor that could then be used to outline a generalized homology thought. developing the periodicity of order , the writer indicates, following Atiyah, concrete cognizance of K-homology is obtained.

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We begin with injectivity. As we said above, standard arguments show that Ext_ 1 (X) rally isomorphic to the subgroup of Ext (XxS 2 ) =Ext(S2 X) is natu- of elements which map to 0 in both Ext (X) and Ext (S 2 ). ) Let p: Xx S2 --. X be the projection. We want to show that for a in Ext(XxS2 ) such that p*(a) = 0 and Per*(a) = 0, it follows that a= 0. If we let L denote the pullback of the Hopf bundle to Xx s2 , then one can show that Hence we want to show that for a in Ext(XxS 2 ) satisfying p*(a) = p*([L] ·a) = 0, it follows that a = 0.

30 c* -ALGEBRA EXTENSIONS AND K-HOMOLOGY Recall that the inverse limit of a sequence of spaces {Xn I and maps fn: xn+l ... : 1) is defined to be with the subspace topology. : 1) the inverse limit is defined to be One always has the homomorphism P : Ext (lim Xn) .... -> lim Ext (Xn) .... defined by P(r) = I pk*(r)l, where Pk: l~m xn ... xk is the coordinate projection. THEOREM 8. If { (Xn, fn)! is an inverse limit of compact metrizible spaces and continuous maps, then the induced map P ·is surjective.

One way for this to happen is for F to be essentially self-adjoint and Fredholm; then the positive essential spectrum determines an element of Ext (X) while the negative determines its inverse. CHAPTER 3 Ext AS A HOMOTOPY FUNCTOR In the first two chapters we have defined X ~ Ext (X) and shown it to be a covariant functor from the category of compact metrizable spaces to the category of abelian groups. In this chapter we derive the first piece of the exact sequence for Ext and the fact that Ext is a homotopy functor.

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