Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a [positive definite function](http://en.wikipedia.org/wiki/Positive-definite_function_on_a_group) on $G$ using $\xi$?


This question is rather vague and open-ended so I've made this CW, but I'd be *very* appreciative of any references that try to do this.



Positive definite functions on G from Hilbert space vectors?
I'm assuming you mean positive definite in the sense of <http://en.wikipedia.org/wiki/Positive-definite_function_on_a_group>. Well, consider the representation of $G$ on $\ell^2(G)$ by $\left(U\_g v\right)\_h = v\_{g^{-1} h}$. Then you might take the positive definite $L(\ell^2(G))$-valued function $F$ on $G$ defined by $F(s) = P \circ U\_g$ where $P$ is the orthogonal projection on $\xi$.


