# On graph products of multipliers and the Haagerup property for $C^*$-dynamical systems

## TL;DR

This paper explores how the Haagerup property is preserved under graph product constructions of group actions on $C^*$-algebras, introducing new methods involving multipliers and providing broader implications for groupoid theory.

## Contribution

It introduces the graph product of multipliers for actions and proves the preservation of the Haagerup property under these graph products, extending known group results.

## Key findings

- Graph product of multipliers preserves positive definiteness.
- Haagerup property is maintained under suitable graph product actions.
- Provides an alternative proof for coarse embeddability results.

## Abstract

We consider the notion of the graph product of actions of groups $\left\{G_v\right\}$ on a $C^*$-algebra $\mathcal{A}$ and show that under suitable commutativity conditions the graph product action $\bigstar_\Gamma \alpha_v: \bigstar_\Gamma G_v \curvearrowright \mathcal{A}$ has the Haagerup property if each action $\alpha_v: G_v \curvearrowright \mathcal{A}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive definite multipliers is positive definite. These results have impacts on left transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.