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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.

edited May 25 at 22:41

Jon Bannon

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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..... – S. Sra May 25 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 $g \mapsto \\\|\xi\\\|$? – robot May 25 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 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?) – robot May 25 at 23:00
	@robot: you seem to have typed a few double backslashes because of which this have gotten the undesired linebreaks. – S. Sra May 26 at 0:15

Robert Israel · Accepted Answer · 2012-05-25 18:00:12Z UTC

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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 $\xi$.

answered May 25 at 18:00

Robert Israel

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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 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 at 23:10

Positive definite functions on G from Hilbert space vectors?

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