A GNS - like *-representation of a \pa\ \A defined by certain representable
linear functionals on \A is constructed. The study of the interplay with the
GNS construction associated with invariant positive sesquilinear forms (ips)
leads to the notions of pre-core and of singular form. It is shown that a
positive sesquilinear form with pre-core always decomposes into the sum of an
ips form and a singular one
