Exercise 6, § 3

Let G be a discrete topological group. Let A be the involutive subalgebra of $\mathscr{L}(\ell^2(G))$ generated by the image of the left regular representation $\boldsymbol{\lambda }$. It is a unital star-algebra.

For every positive type function $\varphi \in$ Pos(G), there exists a unique totally positive linear mapping $u_{\varphi}$ of A into A (cf. Exercise 4) such that $u_{\varphi}(\boldsymbol{\lambda }(g)) =\varphi (g)\boldsymbol{\lambda }(g)$ for every $g\in G$. (Use a cyclic Hilbert representation of $\varphi .$)