Let $\mathbb R_+$ be the semi-field of non-negative real numbers.
Definition (preliminary): A Hilbert space over $\mathbb R_+$ is a pair $(H,P)$, where $H$ is a complex Hilbert space, and $P\subset H$ is a self-dual cone: $$ P=\{\xi \in H \mid \langle \xi,\eta\rangle\in \mathbb R_+\,\,\, \forall \eta \in P\}. $$
Example: Given a von Neumann algebra $A$, its non-commutative $L^2$-space $L^2(A)$ (also called the standard form of $A$) is equipped with a self-dual cone $P:=L^2_+(A)$.
Question (preliminary): Does there exist a reasonable notion of tensor product of Hilbert spaces over $\mathbb R_+$?
The underlying Hilbert space of $(H_1,P_1)\otimes (H_2,P_2)$ should be the tensor product of the underlying Hilbert spaces and, given two von Neumann algebras $A_1$ and $A_2$, the tensor product of $(L^2A_1,L^2_+(A_1))$ with $(L^2A_2,L^2_+(A_2))$ should be $(L^2(A_1\otimes A_2),L^2_+(A_1\otimes A_2))$.
I think that the answer to the above question is negative, and that the definition of a Hilbert space over $\mathbb R_+$ was a bit too simplistic. Let $M_n(\mathbb C)$ denote the algebra of $n\times n$ matrices over $\mathbb C$.
Definition: A Hilbert space over $\mathbb R_+$ consists of a complex Hilbert space $H$, along with a collection of of self-dual cones $P_n\subset H\otimes M_n(\mathbb C)$ for every $n\in\mathbb N$ such that for every completely positive map $f:M_n(\mathbb C)\to M_m(\mathbb C)$ the map $id_H\otimes f$ sends $P_n$ to $P_m$.
Question: Does there now exist a reasonable notion of tensor product of Hilbert spaces over $\mathbb R_+$?
[Added later:
I have a candidate definition for the tensor product of $(H,(P_n)_{n\in\mathbb N})$ with $(K,(Q_n)_{n\in\mathbb N})$. Namely, on the Hilbert space $H\otimes K$, one puts the cones $R_n\subset H\otimes K \otimes M_n(\mathbb C)$ defined as the closed $\mathbb R_+$-linear span of the vectors $(1_{H\otimes K}\otimes f)(\xi\otimes\eta)$ for $\xi\in P_m$, $\eta\in Q_l$ and $f:M_{m l}(\mathbb C)\to M_n(\mathbb C)$ a completely positive map.
I don't know how to prove that the cones $R_n$ are self-dual, and would love to see a proof. ]
