description of functions of conditionally negative type on a group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:08:05Z http://mathoverflow.net/feeds/question/37418 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37418/description-of-functions-of-conditionally-negative-type-on-a-group description of functions of conditionally negative type on a group BigBill 2010-09-01T20:27:20Z 2011-04-18T22:22:15Z <p>Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:</p> <p>1) $\psi(x,x)=0$</p> <p>2) $\psi(y,x)=\psi(x,y)$</p> <p>3) for any elements $x_1,...x_n$ and all real numbers $c_1,...,c_n$, with $c_1+...+c_n=0$, the following inequality holds: $$\sum_{i=1}^{n}\sum_{j=1}^{n} c_ic_j\psi(x_i,x_j)\leq 0.$$</p> <p>Let $G$ be a discrete group. Recall that a function $G\rightarrow \mathbb{R}$ is conditionally of negative type if the kernel $\psi$, defined by $\psi(g,h)= \psi(h^{−1}g)$ is conditionaly of negative type.</p> <blockquote> Does there exist class of discrete groups which admit an explicit description of functions which are conditionaly of negative type? </blockquote> http://mathoverflow.net/questions/37418/description-of-functions-of-conditionally-negative-type-on-a-group/37480#37480 Answer by Keivan Karai for description of functions of conditionally negative type on a group Keivan Karai 2010-09-02T09:54:03Z 2010-09-02T09:54:03Z <p>If you add the condition that Jesse mentioned in the comment above, it is a theorem that such functions are always realized from an affine isometric actions of the group $G$ on a Hilbert space. More precisely, suppose $G$ acts continuously by affine isometries on a Hilbert space $H$. Now, define</p> <p>$$\psi (g)= \| g \cdot x-x \|^2$$</p> <p>for an arbitrary point $x$ in the affine space $H$. First, it is easy to see that such functions are always negative definite. More difficult is to show that any negative definite function can be obtained in this way.</p> <p>You can read more about them in Kazhdan's Property (T) by Bekka, de la Harpe and Valette.</p> http://mathoverflow.net/questions/37418/description-of-functions-of-conditionally-negative-type-on-a-group/62177#62177 Answer by Alain Valette for description of functions of conditionally negative type on a group Alain Valette 2011-04-18T22:22:15Z 2011-04-18T22:22:15Z <p>Sorry for the overdue answer! I think it should be added that, on some classes of groups, there are integral representations of conditionally negative definite functions, known as Levy-Khinchin formulae. E.g, for locally compact abelian groups, see:</p> <p>MR0420769 (54 #8781) Forst, Gunnar The Lévy-Hinčin representation of negative definite functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), no. 4, 313–318.</p>