Every ∗-morphism of a Q
locally m-convex (lmc) ∗-algebra E, in an lmc C∗-algebra F, is continuous. The
same is also true if E is taken to be a Fréchet locally convex ∗-algebra.
Thus, the topology of a Fréchet locally convex C∗-algebra (⇔ Fréchet 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
Pták (e.g. Fréchet) 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
Fréchet 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.
