One realization of spectral theorem for me that we want to make sense "the object $:f(T)$" in von Neumann algebra $M$ where $f$ is bounded measurable function with respect to some measure.
- Classifying seperable representation for commutative $C^*$-algebras but why one need to classify in canonical way?
Essentialy abelian $vN$ algebras isomorphic to $L^\infty(X,\mu)$ is nothing but spectral theorem, still some gaps are there so I am feeling why spectral theorem is so important, can people give more different respectives?
The Spectral Theorem says that any selfadjoint (normal, really) operator is an integral over its spectrum with respect to a projection-valued measure. As you say, this allows one to do bounded Borel functional calculus. This for instance serves to show that von Neumann algebras have many projections; so many that any von Neumann algebra is the norm-closure of the span of its projections. Approximating a selfadjoint operator, in norm, by linear combinations of projections is a basic technique that is used in many many proofs.