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

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.

Keywords:

*none*