ABSTRACT
Roughly speaking, all we did in the previous chapter was the following: we put some interesting approximation sequences into a c• -algebra A and tried to find a family {Wt}ter of homomorphisms from this sequence algebra A into algebras of operators on a Hilbert space having the property that a sequence (An) E A is stable if and only if all associated operators Wt(An) are invertible. Two points are of importance and should be emphasized; the first is that stability is equivalent to the invertibility of 'something', whereas the second is that the 'something' is opemtors on a Hilbert space. This second point is of importance since operators have kernels (which have a dimension), they have ranges (which are sometimes closed), there is a spectral theorem, a Fredholm theory, and a lot of further
208 CHAPTER 5. REPRESENTATION THEORY
ingredients which are not immediately available for elements of a general C*-algebra, and which make it much easier to work with operators rather than with elements of an algebra.
