(See detailed abstract in the article.) We single out the correct class of
spatial product systems (and the spatial endomorphism semigroups with which the
product systems are associated) that allows the most far reaching analogy in
their classifiaction when compared with Arveson systems. The main differences
are that mere existence of a unit is not it sufficient: The unit must be
CENTRAL. And the tensor product under which the index is additive is not
available for product systems of Hilbert modules. It must be replaced by a new
product that even for Arveson systems need not coincide with the tensor
product