By Ronald G. Douglas

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

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Relative to weak equivalence. If r: C(X)-> ~(~) is a *-monomorphism, then tensoring by Mn yields Ill Ill 4. GENERALIZED HOMOLOGY THEORY AND PERIODICITY 43 and hence a homomorphism Ext (X) .... Ext (Cxxcn(X)) is defined which can be shown without much difficulty to be an isomorphism ([22], p. 292). <:2(~) to £(~). This can always be done with a subspace of dimension less than n left over-hence the need for weak equivalence. In [66] Olsen and Zame prove that if X C en, then C(X)®Mn has a single generator.

Of Ext(XxS(l-r)) for r < 1, while K-m(X) is naturally J - m isomorphic to the subgroup j~l ker qj of K0 (XxS(m)). If 0~ and ~~ denote the projection maps 4. GENERALIZED HOMOLOGY THEORY AND PERIODICITY ®~: xx s

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].