Given a group $G$, we can form its complex group ring/algebra $\Bbb{C}[G]$, which either be described as the set of all finite formal sums of group elements with complex coefficients or as all the functions $G \to \Bbb{C}$ with finite support. Question:
Given a function $f : G \to \Bbb{C}$ of finite support, what does it mean to say that $f$ is of positive type? And given that description, is it possible to work out what means when we view $f$ as a finite formal sum of group elements with complex coefficients; i.e., what does that say about the group elements in the sum and what does it say about the complex coefficients?
For reference, see the definition of $\Bbb{k}_r[\Gamma]_+$ on page 4 of this.