By Ronald G. Douglas
Recent advancements in assorted parts of arithmetic recommend the research of a undeniable type of extensions of C*-algebras. right here, Ronald Douglas makes use of equipment from homological algebra to review this selection of extensions. He first exhibits that equivalence sessions of the extensions of the compact metrizable house X shape an abelian crew Ext (X). moment, he exhibits that the correspondence X ? Ext (X) defines a homotopy invariant covariant functor which may then be used to outline a generalized homology thought. constructing the periodicity of order , the writer indicates, following Atiyah, concrete awareness of K-homology is obtained.
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Additional resources for C*-Algebra Extensions and K-Homology
If M is the Mobius band, C is the circle running down the center of M, p: M .... M/aM is the quotient map, a: C(M) .... ~(~) is a *-monomorphism, and > is a homeomorphism from s1 onto C , then r can be realized by defining where T g is the Toeplitz operator with symbol g. The proof follows by considering the sequence Ext(aM) ---Ext(M) ---Ext(M/aM) z-----z where Ext (M) can be identified with Ext (C) = of M , and the map from aM .... C is z .... z 2 . Z since C is a retract Therefore Z .... Z is the map n ....
Dimensional real vector bundle over X , then E is said to be oriented for K-theory if there exists a in Kn(S(E)), where S(E) denotes the Sn-bundle in E over X and p': S(E) .... X, which restricts to a generator of Kn(p'- 1 (x)) for each x in X. The element a is said to be a K-theory orientation for E and is not unique, in general. Using standard arguments it is possible to establish a Thom isomorphism theorem (, p. 78). 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)) ....
The difficulty in this approach lies in finding an appropriate Z . One method is contained in the following LEMMA. , ~(~) be such that p*(r) = r(p*) = rroa, and let E( ·) be the spectral measure on Y satisfying a(f) = f fdE for f in C(Y). y If c is a closed subset of y such that ac contains no point with multiple preimage, then rr(E(C)) commutes with im r. Moreover, the maximal ideal space X of the algebra Z generated by im r and rr(E(C)) is p- 1 (C) v p- 1 (Y\ C). The proof is straightforward once one observes that the functions f in C(X) of the form f·= f 1 op +f 2 +f3 , where f 1 is in C(Y), f 2 is 5 Tue topologist will note the analogy (with interesting contrasts) with the splitting principle for vector bundles.