Negative definite functions for C*-dynamical systems

TL;DR

This paper introduces a new concept of negative definiteness for functions from a group to a C*-algebra under an action, extending classical theorems and characterizing the Haagerup property for such actions.

Contribution

It defines $ ext{alpha}$-negative definiteness for functions into C*-algebras and generalizes classical theorems to this setting, providing new insights into group actions.

Findings

Established analogs of Delorme-Guichardet and Schoenberg theorems for C*-algebra-valued functions.

Provided a characterization of the Haagerup property for group actions on C*-algebras.

Extended classical harmonic analysis results to the setting of C*-dynamical systems.

Abstract

Given an action $\alpha$ of a discrete group $G$ on a unital C*-algebra $A$, we introduce a natural concept of $\alpha$-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 $\alpha$ when $G$ is countable.