  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.
