# Loynes spaces, also called pseudo-Hilbert spaces

Let me first define my object:

First, a locally convex space $Z$ is called admissible in the sense of Loynes if

1. $Z$ is complete

2. 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.

3. 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 $(x+y)^* = x^* + y^*$.

4. The topology of $Z$ is compatible with the order.

5. 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:

1. $X$ is endowed with a $Z$-valued inner product (also called gramian), that is there exist a map $X \times X \to Z$, denoted $[x,y] with the properties: 2.$[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$is complete with its topology it is called a Loynes space. In the literature it is also seen the names pseudo-Hilbert. Loynes himself originaly used the names VE-space and VH-space, see the paper R. M. Loynes: "On generalized positive-definite functions", 1964, Proc. London Mathemathical Soc. (3) 15 373-84 and references therein. One use of Loynes space is to define a more general version of stationary stochastic process, with a corresponding Spectral theory. So my question: What have been the effect of the introduction of Loynes spaces, I am especially interested in work using these concepts in probability theory. Comments, and especially references to papers or books are very welcome! - I don't know anything about them, but I presume in your definition that you want$[\cdot,\cdot]$to generate the topology on$X\$ or something? – Nate Eldredge Jun 23 '10 at 2:54
I ditto Nate's question. Also, if Z actually is a C*-algebra, then this concept is nothing but that of a "Hilbert C*-module", about which a lot is known (see e.g. Chris Lance's book, or the K-Theory book by Wegge-Olsen, etc. etc.) – Matthew Daws Jun 23 '10 at 6:49