Representations of Hermitian Commutative ∗-algebras by Unbounded Operators

We give a spectral theorem for unital representions of Hermitian commutative unital ∗-algebras by possibly unbounded operators in a pre-Hilbert space. A more general result is known for the case in which the ∗-algebra is countably generated. 1. Statement of the Main Result Our main result is the following: Theorem 1. Let π be a unital representation of a Hermitian commutative unital ∗-algebra A in a pre-Hilbert space H 6= {0}. The operators π(a) (a ∈ A) then are essentially normal in the completion of H. The spectrum sp(π) ⊂ ∆(A) = ∆(A) of the representation π is defined by sp(π) := { τ ◦ π ∈ CA : τ ∈ ∆ ( π(A) )} . There exist a σ-algebra E on sp(π), and a spectral measure P defined on E and acting on the completion of H, with the following properties: (i) the functions â | sp(π) are E-measurable for all a ∈ A, (ii) for each a ∈ A, the closure π(a) equals ∫ sp(π) â | sp(π) dP , (iii) for each a ∈ A, the spectral resolution of the unbounded normal operator π(a) is the image â | sp(π)(P ) of P under â | sp(π), (iv) a bounded operator b commutes with the unbounded normal operators π(a) (a ∈ A) if and only if b commutes with P . In particular, the normal operators π(a) (a ∈ A) commute spectrally. For the case in which the ∗-algebra is countably generated, the above result is essentially contained in a more general result of Savchuk and Schmüdgen [7, Theorem 7 p. 46]. The notation and terminology is explained in the following section. The proof will be given in section 5 below. Date: 2009, August 22. 2000 Mathematics Subject Classification. Primary: 47A67; Secondary: 47L60, 47C10.