3 - Splitting and the Mayer–Vietoris Sequence

Summary

In this chapter, we address the question of decomposing the semigroup Ext(X) provided a decomposition of X is given. In particular, we introduce the notion of splitting of an extension, and note that a closed disjoint cover ﹛A, B﹜ of X yields direct sum decomposition of Ext(X) into Ext(A) and Ext(B). As the first major step in the ultimate splitting lemma (to be proved in the next chapter), we establish the first splitting lemma, which states that such a decomposition holds for a closed cover ﹛A, B﹜ if A ∩ B is a singleton.

Splitting

Let X be a compact Hausdorff space. Given a*-monomorphism and, write T^f for any operator in such that. It will always be understood that T^f is determined only up to simultaneous unitary equivalence modulo the compacts. Recall that Imτ stands for the image of. If T is in and E is an orthogonal projection in, then write T_E for the operator in.

Lemma 3.1

Let e be an orthogonal projection in the Calkln algebra and τ_e : be a *-monomorphism such that τ_e(1) = e. If E is an orthogonal projection in such that π(E) =e (see Corollary 2.2.1), then we have the following statements:

• (1) (Existence) There exists a unital *-monomorphism such that

• where and.

• (2) (Uniqueness) If F is another orthogonal projection such that π(F) = e, then τ_e,E is equivalent to τ_e,F.

Proof (1) Note that implies that, that is,. Thus, the map is well defined. Similarly, one can see that the projection e commutes with, and hence,

If we decompose the operator T^f with respect to E and I − E, then by (3.1.1), the off diagonal entries are compact. Thus, the map is *-homomorphism.

(2) Let F be an orthogonal projection in such that and note that E − F is a compact operator. Let U and V be isometries in such that and are unitaries (see Corollary 1.5.1). Define and by

Since EU and FV are unitaries,

As weakly equivalent extensions are equivalent (see Proposition 2.5.1), it suffices to show that is weakly equivalent to Observe that

In the last but one equality, we have used the fact that E and F differ by a compact operator.