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

Abstract: 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 $\G$ arises as a transform of a positive functional on the universal C*-algebra $C_0^u(\dual\G)$ of the dual quantum group. Second, when $\G$ 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 $\lone_\sharp(\G)$, and when $\G$ is coamenable, the Banach *-algebra $\lone_\sharp(\G)$ has a contractive bounded approximate identity.