Positive-definite function on a group

Remarks

Equivalently, each matrix $\big(\varphi(g_i^{-1}g_j)\big)_{i,j=1}^{n}$ is Hermitian and positive semidefinite. It follows that $\varphi(e) \ge 0$, $\varphi(g^{-1}) = \overline{\varphi(g)}$ and $\abs{\varphi(g)} \le \varphi(e)$ for all $g$. For a discrete group, or when continuity is not part of the setting, the same condition defines positive-definite functions without the continuity requirement.

Example: let $\pi$ be a unitary representation of $G$ on $H$ and $v \in H$, and put $\varphi(g) = \ip{\pi(g)v}{v}$. Then $\varphi$ is continuous, and $$\sum_{i,j} \overline{c_i} c_j \ip{\pi(g_i^{-1}g_j)v}{v} = \sum_{i,j} \overline{c_i} c_j \ip{\pi(g_j)v}{\pi(g_i)v} = \Big\lVert \sum_j c_j \pi(g_j)v \Big\rVert^2 \ge 0.$$

Non-example: the constant function $\varphi \equiv -1$. With $n = 1$ and $c_1 = 1$, the sum is $\varphi(e) = -1 < 0$.

Conversely, every continuous positive-definite function has the form $g \mapsto \ip{\pi(g)v}{v}$ for some unitary representation $\pi$ and a cyclic vector $v$, by a construction of Gelfand–Naimark–Segal type.