Title:The CCAP for graph products of operator algebras

Abstract:For a simple graph $\Gamma$ and for unital $C^*$-algebras with GNS-faithful states $(\mathbf{A}_v,\varphi_v)$ for $v\in V\Gamma$, we consider the reduced graph product $(\mathcal{A},\varphi)=*_{v,\Gamma}(\mathbf{A}_{v},\varphi_v)$ , and show that if every $C^*$-algebra $\mathbf{A}_{v}$ has the completely contractive approximation property (CCAP) and satisfies some additional condition, then the graph product has the CCAP as well. The additional condition imposed is satisfied in natural cases, for example for the reduced group $C^*$-algebra of a discrete group $G$ that possesses the CCAP.
Our result is an extension of the result of Ricard and Xu in [Proposition 4.11, 25] where they prove this result under the same conditions for free products. Moreover, our result also extends the result of Reckwerdt in [Theorem 5.5, 24], where he proved for groups that weak amenability with Cowling-Haagerup constant $1$ is preserved under graph products. Our result further covers many new cases coming from Hecke-algebras and discrete quantum groups.