Some remarks on the GNS representations of topological $^*$-algebras

TL;DR

This paper revisits the GNS construction for topological $^*$-algebras, establishing isomorphisms among various representation spaces and exploring their implications for cyclic representations.

Contribution

It introduces an isomorphism framework linking weakly continuous cyclic representations, Hilbert spaces embedded in the dual space, and reproducing kernels for topological $^*$-algebras.

Findings

Established isomorphisms as cone morphisms among representation sets

Connected cyclic representations with reproducing kernel Hilbert spaces

Provided insights for describing cyclic representations on general inner product spaces

Abstract

After an appropriate restatement of the GNS 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 $\hilb_A(A^*)$ 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.