Commutative von Neumann algebras and representations of normal Hilbert space operators, or “Spectral Measures without the Spectrum” (title approved by Dr Leader)

One can view a complex Hilbert space as the natural infinite dimensional analog of the spaces C, and many of the most fundamental intuitions about Hilbert space geometry rest on a direct analogy with finite-dimensional geometry. While this intuition is certainly the right one for the basic foundations of the theory of Hilbert spaces, we should perhaps take ourselves to task about it: why should there be a sensible infinite-dimensional analog of the geometry of C? Perhaps more importantly, why does such a thing matter? In fact, Hilbert spaces are special precisely because of this direct generalization of finite dimensional geometry. Arguably they are the most artificial of all Banach spaces. While an M-space, for example, has a vast range of possible substructure, a Hilbert space has a geometric structure so homogeneous that even its own isometries cannot tell one point of the unit sphere from another. This absence of pathologies results in a highly agreeable geometry, and it is this itself, and the related behaviour of other constructions based on it, that makes Hilbert spaces worthy of study. However, by itself the maxim that we should be able to extend finite-dimensional thinking will not get us very far. We need naturally-occurring Hilbert spaces to facilitate our study; and we find them in integration theory as LC spaces (including, in particular, the space ` 2 C, when we consider counting measure on the natural numbers). (Note that we will persist in writing the subscript C in acknowledgement of the convention that L is a real space.) In fact, we can say more: for many purposes, these are the only naturally-occurring Hilbert spaces that matter (an arguable exception to this is the Hardy space H, but in comparison this is of importance only under very special circumstances: the study of shift operators; see [11] for a comprehensive treatment). A central message of this essay is that additional structures on a Hilbert space are often best understood by representing that Hilbert space itself in the form of one of these old friends. This can shed more light on the situation than a representation based only on higher-level structures such as algebras of operators. As Masamichi Takesaki put it in [15],