Some remarks on the Gelfand–Naimark–Segal representations of topological *-algebras

After an appropriate restatement of the Gelfand–Naimark–Segal construction for topological *-algebras we prove that there exists an isomorphism among the set Cycl(A) of weakly continuous strongly cyclic *-representations of a barreled dual-separable *-algebra with unit A, the space HilbA(A*) of the Hilbert spaces that are continuously embedded in A* and are * -invariant under the dual left regular action of A, and the set of the corresponding reproducing kernels. We show that these isomorphisms are cone morphisms and we prove many interesting results that follow from this fact. We discuss how these results can be used to describe cyclic representations on more general inner product spaces.