Spectral properties of Abelian \(C^*\)-algebras (Q1579890)

Let \({\mathcal A}\) be an Abelian unital \(C^*\)-algebra, \(\widehat{\mathcal A}\) its Gelfand spectrum and \((H,\pi)\) a nondegenerate representation of \({\mathcal A}\). Let \(H_x\) be the cyclic \(\pi({\mathcal A})\)-invariant subspace of \(H\) given by the closure of \(\{\pi(A)x\mid A\in{\mathcal A}\}\); \(\mu_x\) be the positive Baire measure on \(\widehat{\mathcal A}\), where \(x\in H\). In this paper some necessary and sufficient conditions are proved for a nondegenerate representation of \({\mathcal A}\) to be unitarily equivalent to a representation in which the elements of \({\mathcal A}\) act multiplicatively, by their Gelfand transforms, on a space \(L^2(\widehat{\mathcal A},\mu)\). Namely, the author proved the equivalence of the following three properties: (i) the projector on \(H_x\) belongs to the Baire *-algebra generated by \(\pi({\mathcal A})\); (ii) there exists a Baire subset \(S_x\) of \(\widehat{\mathcal A}\) such that \(\mu_x(\widehat{\mathcal A}\setminus S_x)= 0\) and \(\mu_y(S_x)= 0\), where \(y\perp H_x\); (iii) there exist a positive measure \(\mu\) on the Baire sets of \(\widehat{\mathcal A}\) and an isometric isomorphism \(U: H\to L^2(\widehat{\mathcal A},\mu)\) such that \(U\pi(A) U^{-1}\) is the operator of multiplication by the Gelfand transform of \(A\), for all \(A\in{\mathcal A}\). Moreover, he compares these conditions with the multiplicity-free property of a representation.