The notion of observable and the moment problem for *-algebras and their GNS representations

Nicolò Drago, Valter Moretti
March 18, 2019
Dealing with $^*$-algebras $A$ (not $C^*$-algebras) the notion of observable is delicate. It is generally false that for $a=a^* \in A$ the operator $\pi_\omega(a)$ in a GNS rep. of a state $\omega$ is essentially selfadjoint: it is symmetric admitting many or none selfadjoint extensions. The problem is entangled with the physical meaning of $\omega(a)$ as expectation value. It needs a probability measure $\mu_{\omega}^{(a)}$ arising from the PVM of $\overline{\pi_\omega(a)}$ if it is selfadjoint. The problem of finding $\mu_\omega^{(a)}$ can be also tackled in the framework of the Hamburger moment problem, looking for a probability measure with moments $\omega(a^n)$ for $n=0,1,2, \ldots$. However, for a $^*$-algebra which is not $C^*$, there are many such measures for given $(a,\omega)$ no matter if $\pi_\omega(a)$ admits one many or none selfadjoint extensions. These issues are studied focusing on the information provided by states $A \ni c \mapsto \omega_b(c) := \omega(b^* c b)/\omega(b^*b)$, with $b\in A$. The solutions of the moment problem $\mu_{\omega_b}^{(a)}$ for moments $\omega_b(a^n)$ is analyzed. We prove that, if the measures $\mu_{\omega_b}^{(a)}$ are uniquely determined by $b$ and $(a, \omega)$, then $\overline{\pi_{\omega_b}(a)}$ are selfadjoint. The converse is false. Furhtermore, for fixed $a^*=a\in A$ and $\omega$, under natural coherence constraints on $\mu_{\omega_b}^{(a)}$, the admitted families of measures $\{\mu^{(a)}_{\omega_b}\}_{b\in A}$ are one-to-one with all POVMs associated to the symmetric operator $\pi_\omega(a)$ through Naimark's theorem. $\overline{\pi_\omega(a)}$ is maximally symmetric iff such measures are unique for every $b$. These measures are induced by the unique POVM of $\overline{\pi_\omega(a)}$, which is a PVM if the operator is selfadjoint.