---
tag: 1J67Y01N4X
kind: definition
name: Positive-definite function on a group
msc: ["43A35", "22D10"]
defines: 3PRZ02APFA
uses: [92700QAB5Y, 0KX66CZMBB, 74BPYJNZTM]
sources:
  - url: https://en.wikipedia.org/wiki/Positive-definite_function_on_a_group
---

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] with identity $e$.
A continuous function $\varphi \colon G \to \C$ is *positive-definite* if for every $n \ge 1$, all $g_1, \dots, g_n \in G$ and all $c_1, \dots, c_n \in \C$,
$$\sum_{i=1}^{n} \sum_{j=1}^{n} \overline{c_i} \, c_j \, \varphi(g_i^{-1} g_j) \ge 0.$$

## Remarks

Equivalently, each matrix $\big(\varphi(g_i^{-1}g_j)\big)_{i,j=1}^{n}$ is Hermitian and [[positive semidefinite|0KX66CZMBB]].
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|74BPYJNZTM]] 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.