turkmath.org

An ordered *-space Z is a complex vector space with a conjugate linear involution *, and a strict cone Z+ consisting of self-adjoint elements. We discuss ordered *-spaces with a suitable locally convex topology, called topologically ordered *-spaces. Then we talk about complex vector spaces which have an "inner product" valued in a complete topologically ordered *-space, called VH (Vector Hilbert) spaces (in the sense of Loynes) in case their induced locally convex topology is complete, and their linear operators. Examples of all these spaces will also be discussed. This talk is the first of a series of planned talks on a general dilation theorem in the context of VH-spaces, and its applications.