Math 241b Functional Analysis - Notes Spectral Theorem

We present the material in a slightly different order than it is usually done (such as e.g. in the course book). Here we prefer to start out with an abelian C∗-algebra A (say, the algebra C∗(a) generated by a normal operator a ∈ B(H)) and construct from it the spectral measure. This is done as follows: We know that A is isomorphic to C(X), the algebra of continuous functions on a compact set X. So we can view our concrete algebra A ⊂ B(H) as the image of a representation π : C(X) → B(H). We first show that this representation can be extended to a representation, also denoted by π of the bounded Borel functions B(X) into B(H). We denote by χU the characteristic function of a measurable subset U ⊂ X, i.e. χU (x) is either 1 or 0, depending on whether x ∈ U or not. Then the spectral measure E(U) is the projection given by E(U) = π(χU ), U ⊂ X measurable. If A = C∗(a), for a a normal operator, and X = σ(a) its spectrum, we obtain an analog of the eigenspace decomposition of a normal matrix, with the spectral spaces E(U)H corresponding to direct sums of eigenspaces in the finite dimensional case. In order to work out the details, we need a few basic results. For the definition of regular measure, see e.g. the course book, Appendix C.10, or wikipedia.