  We introduce the notion of additive units and roots of a unit in a spatial
product system. The set of all roots of any unit forms a Hilbert space and its
dimension is the same as the index of the product system. We show that a unit
and all of its roots generate the type I part of the product system. Using
properties of roots, we also provide an alternative proof of the Powers'
problem that the cocycle conjugacy class of Powers sum is independent of the
choice of intertwining isometries. In the last section, we introduce the notion
of cluster of a product subsystem and establish its connection with random sets
in the sense of Tsirelson and Liebscher.

Functional Analysis