Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 point wise.
Again the positive definite-ness of some function $\phi:G \rightarrow \mathbb{R}$ says that there is a real Hilbert spaces $\mathcal{H}$ with an orthogonal representations $$\pi:G \rightarrow \mathcal{H}$$ with the property that $\phi(g)=\langle\pi(g)\xi,\xi\rangle$, for some vector $\xi \in \mathcal{H}$ and all $g\in G$.
I know that free product of two amenable groups is not amenable, but I also know that it has Haagerup property, i.e. on this free product we have a sequence of positive definite functions that are vanishing at infinity and converges to 1 point wise.
I know that this can be shown by the existence of a proper conditionally negative definite function and Schoenberg lemma.
But my question is can one construct explicitly a orthogonal representation of the free product $\Gamma=A\ast B$ using the orthogonal representations of $A$ and $B$, that come along the definition of amenability on that groups.
Thanks in advance.
