Completely positive definite functions and Bochner's theorem for locally compact quantum groups

We prove two versions of Bochner’s theorem for locally compact quantum groups. First, every completely positive definite “function” on a locally compact quantum group $\text{G}$ arises as a transform of a positive functional on the universal C*-algebra $C_0^u(\text{dual}\text{G})$ of the dual quantum group. Second, when $\text{G}$ is coamenable, complete positive definiteness may be replaced with the weaker notion of positive definiteness, which models the classical notion. A counterexample is given to show that the latter result is not true in general. To prove these results, we show two auxiliary results of independent interest: products are linearly dense in ${\text{lone}}_{\mathrm{♯}}(\text{G})$, and when $\text{G}$ is coamenable, the Banach *-algebra ${\text{lone}}_{\mathrm{♯}}(\text{G})$ has a contractive bounded approximate identity.