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

TL;DR

This paper extends Bochner's theorem to locally compact quantum groups, characterizing positive definite functions as transforms of positive functionals, with special results for coamenable quantum groups.

Contribution

It proves two versions of Bochner's theorem for quantum groups, linking positive definiteness to positive functionals on universal C*-algebras, and explores properties of associated Banach *-algebras.

Findings

Complete positive definiteness corresponds to transforms of positive functionals.

When quantum groups are coamenable, positive definiteness suffices, unlike in the general case.

Key auxiliary results include density of products in $ ext{L}^1_lat( ext{G})$ and existence of bounded approximate identities.

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.