Negative definite functions for C*-dynamical systems

Abstract

Given an action $α$ of a discrete group $G$ on a unital C*-algebra $A$, we introduce a natural concept of $α$-negative definiteness for functions from $G$ to $A$, and examine some of the first consequences of such a notion. In particular, we prove analogs of theorems due to Delorme-Guichardet and Schoenberg in the classical case where $A$ is trivial. We also give a characterization of the Haagerup property for the action $α$ when $G$ is countable.