  We consider the algebra $\mathcal O(\mathsf M)$ of observables and the
(formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the
noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family
$\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative
$k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is
an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of
simple $A$-modules, and $k$ is an algebraically closed field. The purpose of
this paper is twofold: First, we prove a form of the Generalized Burnside
Theorem that is more general, where there is no assumption on the field $k$.
Secondly, we prove that the $\mathcal O$-construction is a closure operation
when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of
finite dimensional $A$-modules, in the sense that $\eta_B: B \to \mathcal
O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf
M$ is considered as a family of $B$-modules.
