Conditionally negative definite function

Summary

A conditionally negative definite function on a group $G$ is a real function $\psi$ with $\psi(e) = 0$ and $\psi(g^{-1}) = \psi(g)$ and one more condition. The sums $\sum_{i,j} c_i c_j \psi(g_j^{-1} g_i)$ are $\le 0$ whenever the real coefficients $c_i$ add up to $0$. The typical example is $\psi(g) = \norm{b(g)}^2$, the squared displacement of a point under an affine isometric action on a real Hilbert space. By Schoenberg's theorem, $\psi$ is conditionally negative definite exactly when $e^{-t\psi}$ is a positive-definite function for every $t > 0$. These functions measure how a group can act on Hilbert spaces. They appear in the theory of property (T), of the Haagerup property, and of Lévy processes.