Every *-morphism of a Q locally ra-convex (lmc) *-algebra E, in an lmc C* -algebra F, is continuous. The same is also true if E is taken to be a Frechet locally convex * -algebra. Thus, the topology of a Frechet locally convex C*-algebra (& Frechet lmc C*-algebra) is uniquely determined. Each lmc C* -algebra has a continuous involution. In the general case, one has that the involution of a barrelled Ptak (e.g. Frechet) locally convex algebra E is continuous iff the real locally convex space H{E) of its self-adjoint elements, is a closed subspace. In particular, every algebra E as before, which admits a continuous faithful * -representation, has a continuous involution. Furthermore (without assuming continuity of the involution), we obtain that every * -representation of an involutive Frechet Q lmc algebra E, is continuous, while if E has moreover a bounded approximate identity, the same holds also true for each positive linear form of E.
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