  It is now well established that quantum tomography provides an alternative
picture of quantum mechanics. It is common to introduce tomographic concepts
starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert
spaces. In this picture states are a primary concept and observables are
derived from them. On the other hand, the Heisenberg picture,which has evolved
in the $C^\star-$algebraic approach to quantum mechanics, starts with the
algebra of observables and introduce states as a derived concept. The
equivalence between these two pictures amounts essentially, to the
Gelfand-Naimark-Segal construction. In this construction, the abstract $%
C^\star-$algebra is realized as an algebra of operators acting on a constructed
Hilbert space. The representation one defines may be reducible or irreducible,
but in either case it allows to identify an unitary group associated with the
$C^\star-$algebra by means of its invertible elements. In this picture both
states and observables are appropriate functions on the group, it follows that
also quantum tomograms are strictly related with appropriate functions
(positive-type)on the group. In this paper we present, by means of very simple
examples, the tomographic description emerging from the set of ideas connected
with the $C^\star-$algebra picture of quantum mechanics. In particular, the
tomographic probability distributions are introduced for finite and compact
groups and an autonomous criterion to recognize a given probability
distribution as a tomogram of quantum state is formulated.
