Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\not = 0)$ that $x \in Z_+$ implies $-x \not\in Z_+$, which is used to define a partial order relation on $Z$ in the usual way.
There is an involution in $Z$ (which in the real case can be taken to be identity), $(x^*)^*=x$, $(\alpha z)^* = \bar{\alpha} z^*$ and finally $(x+y)^* = x^* + y^*$.
The topology of $Z$ is compatible with the order.
Any monotonously decreasing sequence in $Z_+$ is convergent.
An easy example for $Z$ could be a $C^*$-algebra.
Then let $Z$ be an admissible space in the sense of Loynes. A linear topological space $X$ is called pre-Loynes if it satisfies:
$X$ is endowed with a $Z$-valued inner product (also called gramian), that is there exist a map $X \times X \to Z$, denoted $[x,y] with the properties:
$[x,x]\ge 0$, $[x,x]=0$ only if $x=0$. The usual bilinear properties and finally, $[x,y]^*=[y,x]$, that is, in the real case, symmetry.
There are also some topological conditions. If the space $X$ is complete with its topology it is called a Loynes space. In the literature it is also seen the names pseudo-Hilbert. Loynes himself originaly used the names VE-space and VH-space, see the paper R. M. Loynes: "On generalized positive-definite functions", 1964, Proc. London Mathemathical Soc. (3) 15 373-84 and references therein.
One use of Loynes space is to define a more general version of stationary stochastic process, with a corresponding Spectral theory.
So my question: What have been the effect of the introduction of Loynes spaces, I am especially interested in work using these concepts in probability theory. Comments, and especially references to papers or books are very welcome!