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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 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.

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Can one define heat kernels here, as in solutions to Laplace-Baltrami diff. equations? if so, that would be the "most natural" way to define at least one posdef function..... –  Suvrit May 25 '12 at 16:11
    
$\xi \overline{\xi}$? Or do you want a function $f: G\to \mathbb{R}$ such that $f(g)>0$ for all $g\in G$? What about constant function $g \mapsto \\|\xi\\|$? –  Vít Tuček May 25 '12 at 16:30
    
@robot: I've edited the question to include a link to what I mean by positive definite function. –  Jon Bannon May 25 '12 at 22:43
    
Thanks. Now where do I report that bug with math display? (Or is it just me who sees \xi directly below g in my previous comment?) –  Vít Tuček May 25 '12 at 23:00
    
@robot: you seem to have typed a few double backslashes because of which this have gotten the undesired linebreaks. –  Suvrit May 26 '12 at 0:15

1 Answer 1

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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 $\xi$.

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Thanks, Robert. Indeed this example is intrinsic to the vector as I asked. Perhaps composition of this with any state would give me scalar-valued positive definite functions, as well... –  Jon Bannon May 25 '12 at 22:42
    
Actually, Robert, this sort of p.d. function seemst to give me exactly what I need. I'll accept this answer in a few days if no other contributions appear. Thanks again! –  Jon Bannon May 25 '12 at 23:10

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