Operator approximate biprojectivity of locally compact quantum groups

Abstract

We initiate a study of operator approximate biprojectivity for quantum group algebra $L^1\left(\mathbb{G}\right)$, where $\mathbb{G}$ is a locally compact quantum group. We show that if $L^1\left(\mathbb{G}\right)$ is operator approximately biprojective, then $\mathbb{G}$ is compact. We prove that if $\mathbb{G}$ is a compact quantum group and $\mathbb{H}$ is a non-Kac-type compact quantum group such that both $L^1\left(\mathbb{G}\right)$ and $L^1\left(\mathbb{H}\right)$ are operator approximately biprojective, then $L^1\left(\mathbb{G}\right)\hat{\otimes }{L}^1\left(\mathbb{H}\right)$ is operator approximately biprojective, but not operator biprojective.