Fully representable and *-semisimple topological partial *-algebras

We continue our study of topological partial *-algebras, focusingour attention to *-semisimple partial *-algebras, that is, those that possessa multiplication core and su ciently many *-representations. We discuss therespective roles of invariant positive sesquilinear (ips) forms and representablecontinuous linear functionals and focus on the case where the two notions arecompletely interchangeable (fully representable partial *-algebras) with thescope of characterizing a *-semisimple partial *-algebra. Finally we describevarious notions of bounded elements in such a partial *-algebra, in particular,those defined in terms of a positive cone (order bounded elements). The outcomeis that, for an appropriate order relation, one recovers the M-boundedelements introduced in previous works.