By Ronald G. Douglas

Recent advancements in varied parts of arithmetic recommend the learn of a undeniable category of extensions of C*-algebras. right here, Ronald Douglas makes use of equipment from homological algebra to review this selection of extensions. He first indicates that equivalence periods of the extensions of the compact metrizable house X shape an abelian staff Ext (X). moment, he indicates that the correspondence X ? Ext (X) defines a homotopy invariant covariant functor that can then be used to outline a generalized homology conception. developing the periodicity of order , the writer exhibits, following Atiyah, concrete recognition of K-homology is obtained.

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**Example text**

Then for each n we have oe(ffi ks\ eo+ K' k=l = (EB ~ N\ k=O where K', K" are compact and Te(~: ~ N) x + K"' } I X\I n = I EB } S\I ~ \I S\I . Since = k=l is unitarily equivalent to a compact perturbation of T, it follows that T is in the closure of lies in / Ji + K. (This would imply T itself Ji + K by the result stated in Chapter 1 except our argument would be circular. , j=l i=O Tje 0, where the zero term is infinite dimensional. Since 21 [T, T*] = e[Ti, Ti'J + K' is compact, it follows that \l[Ti, Ti'Jll-> 0 and 00 hence [R, R*] = 2j-1 EB EB -J- [Tj, Tj] j=l i=O is compact.

However, we can define where ry : C(Y) .... ~(~) defines the trivial extension. The properties necessary to be a covariant functor follow easily. Now the most surprising thing about Ext (when we were first studying it and still today) is the fact that Ext(X) is an abelian group. In no situation were algebra extensions known to form a group and we were led to expect it only because Ext (X) was a group for the cases we could calculate. lr original proof of this was quite complicated and came after the construction of most of the other machinery in [20].

THEOREM 13. If p : E .... X is a rank n+ 1 vector bundle over X and a in Kn(S(E)) is a K-theory orientation, then the map c* -ALGEBRA 52 defined by T q(a) = p~(a EXTENSIONS AND K-HOMOLOGY n a) is an isomorphism, where Kq is the kernel of p*: Extq(S(E)) .... Extq(X). The proof proceeds in three steps. First, one considers the case of a product Xx Rn+l, where a comes from a generator of Kn(sn), which follows since r q is a power of Per*. Second, one allows a to be the product of a generator by a unit of K0 (X).