turkmath.org

A VH-space (Vector Hilbert Space in the sense of Loynes) is a complex complete locally convex space with a suitable ordered vector space valued inner product. Examples of VH-spaces include, but is not limited to, the chain of locally Hilbert $C^*$-modules, Hilbert $C^*$-modules and Hilbert Spaces

    We prove that, on a Barreled VH-Space, the set of all adjointable operators consists of bounded operators and is a Locally $C^*$-Algebra, generalizing the well known corresponding fact from the theory of Locally Hilbert $C^*$-modules. 

    We pick a consequence of this result in the dilation theory of VH-Spaces and show that, under the barreledness assumption, a necessary and sufficient condition for the existence of VH-space linearisations, equivalently, of reproducing kernel VH-Spaces, is satisfied automatically.