  We continue our study of topological partial *algebras, focusing our
attention to the interplay between the various partial multiplications. The
special case of partial *-algebras of operators is examined first, in
particular the link between the strong and the weak multiplications, on one
hand, and invariant positive sesquilinear (ips) forms, on the other.
  Then the analysis is extended to abstract topological partial *algebras,
emphasizing the crucial role played by appropriate bounded elements, called
$\M$-bounded. Finally, some remarks are made concerning representations in
terms of the so-called partial GC*-algebras of operators.
