Definition 4

A continuous function $\varphi \in \mathscr{C}(G)$ is said to be of positive type on G if the function $f$ defined by $f(g, h) =\varphi (g^{-1}h)$ on $G\times G$ is a universally positive kernel on G.

Pos(G) denotes the set of functions of positive type on G, and Pos$_1(G) ($resp. Pos$_{\leqslant 1}(G))$ denotes the subset of $\varphi \in$ Pos(G) such that $\varphi (e) = 1 ($resp. such that $\varphi (e)\leqslant 1)$.