The Image of Unitary Representation in the Space of Bimodules 
TL;DR: Bimodules over a von Neumann algebra are commonly understood as
  a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $\mathcal{L} G$, the unitary representations embed into the
  bimodules. 
Question: Can we describe the image of this embedding? Which are the
  bimodules over $\mathcal{L} G$ that come from a representation?

Let $G$ be a discrete group. The four objects in the diagram bellow are:


*

*$\mathrm{Rep_{cyc}}(G)$: its cyclic unitary representations, i.e. unitary representations $\pi: G \to U(H_\pi)$ for which there is a vector $\xi_\pi \in H_\pi$ such that $\{\pi(g) \xi : g \in G\}$ spans a dense subset of $H_\pi$.

*$\mathcal{P}(G)$: The cone of positive definite functions on $G$.

*$\{ {}_N H _N\}$: The space of all $N$-$N$-bimodules, i.e: Hilbert spaces with normal commuting left and right actions of $N$. In this case $N = \mathcal{L} G$, the left regular von Neumann algebra of $G$. We ask the bimodules to be cyclic, i.e.: there has to be a $\xi \in H$ such that $x \cdot \xi \cdot y$ is dense and that $\xi$ is left $N$-bounded.

*$\mathcal{CP}(N \to N)$: The cone of completely positive and normal maps.


All those objects are closely related. For instance, there are bijective correspondences between the objects in each of the columns of the diagram bellow. The objects in the right column are interpreted as "generalizations" of those on the right column from the context of groups to that of von Neumann algebras. The arrows from the left column to the right one are injections.

In particular the maps above are given by:


*

*(i) Given a cyclic representation $\pi$ there is a positive definite function $\varphi$ given by $$\varphi(g) = \langle \xi, \pi(g) \xi \rangle.$$

*(ii) Given a positive definite function $\varphi$ we can obtain a rep. by a GNS-type construction. Define a positive form in $\ell^1(G)$ given by $\langle \varphi, \zeta^\ast \ast \zeta \rangle$. Quotienting out the nullspace and taking completions gives a Hilbert space with a cyclic representation of $G$.

*(iii) Every unitary representation $\pi$ gives rise to a $\mathcal{L} G$-bimodule $\ell^2(G) \otimes H_\pi$ with left and right actions given by:
$$
  \lambda_g \cdot ( \delta_k \otimes \xi ) \cdot \lambda_h = \delta_{g k h} \otimes \pi(g) \xi.
$$

*(iv) If $\varphi$ is positive definite, the Fourier multiplier given by extension of $\lambda_g \mapsto \varphi(g) \lambda_g$ is normal and cp.

*(v) Given a bimodule $H$ with a left $N$-bounded cyclic vector $\xi$ we can construct a normal cp map by $x \mapsto L_\xi^\ast x L_\xi$

*(vi) Given a normal cp map $\phi: N \to N$ we can construct a bimodule $L^2(N \otimes_\varphi N)$ by a GNS-type construction. Take $N \otimes_{alg} N$ and complete it with respect to
$$
  \big\langle x_1 \otimes y_1, x_2 \otimes y_2 \big\rangle = \tau_N \big( x_1^\ast \, \phi(y_1^* y_2) \, x_2 \big).
$$
The actions are given by $x \cdot (a \otimes b) \cdot y = x a \otimes b y$. The construction above requires that $N$ is finite since it uses the trace (no problem since $N = \mathcal L G$).


More on (iii), (v) and (vi) can be found in [AP: Chapter 13].

Question 1: The image of (iv) can be described as the $\Delta$-equivariant cp maps, i.e. maps $T$ such that:
  $$   (T \otimes id) \cdot \Delta = \Delta \cdot T, $$
  where $\Delta: N \to N \otimes N$ is the natural comultiplication $\Delta(\lambda_g) = \lambda_g \otimes \lambda_g$. Can something similar be said of the image of
  (iii)? Is there a notion of "equivariant" bimodule?
Question 2: Given a normal cp map $\phi: \mathcal L G \to \mathcal L G$ we can associate to it a positive definite function $\varphi(g) = \tau(\phi(\lambda_g) \lambda_g^\ast)$. Is there an analogous
  construction that gives an equivariant bimodule from a general one?



*

*[AP]: http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf
 A: I forgot to post one of the possible answers. So with some delay here it goes:
Let $\Delta: \mathcal{L} G \to \mathcal{L} G \otimes \mathcal{L} G$ be the natural comultiplication. We can associate to it a correspondence $H(\Delta) = _{N}H(\Delta)_{N \otimes N} \in \mathrm{Corr}(N, N \otimes N)$, where $N = \mathcal{L} G$ given by $L^2(G \times G)$ with the actions given by
$$
 x \cdot \xi \cdot y = \Delta(x) \xi y.
$$
Similarly, given two correspondences $_NH_M$ and ${}_{M}{K}_R$, there is a module tensor product $_{N} {(H \otimes_{M} K)}_R$, see again (Section 13.2. http://www.math.ucla.edu/~popa/Books/IIun-v13.pdf). Then, the image of the arrow $(iv)$ satisfies that 
$$
  H(\varphi) \otimes_N H(\Delta) = H(\Delta) \otimes_{N \otimes N} \big( L^2(N) \otimes_{\mathbb{C}} H(\varphi) \big),
 \tag{$\star$}
$$
where the equality doesn't mean isomorphism but the fact that the isomorphism is induced by the natural map. A reciprocal follows and the map $\varphi(x) = L^\ast_\xi \, x \, L_\xi$ associated to a correspondence satisfying $(\star)$ is a Fourier multiplier.
