# Complete metric approximation property for $q$-Araki-Woods algebras

## TL;DR

This paper proves that all $q$-Araki-Woods algebras possess the $w^{ ext{*}}$-complete metric approximation property by transferring properties from $q$-Gaussian algebras using ultraproduct techniques.

## Contribution

It establishes the $w^{ ext{*}}$-complete metric approximation property for all $q$-Araki-Woods algebras, extending known results via ultraproduct methods.

## Key findings

- Proves transfer of radial completely bounded multipliers to $q$-Araki-Woods algebras.
- Establishes the $w^{ ext{*}}$-complete metric approximation property for these algebras.
- Shows canonical dense C$^ ext{*}$-subalgebras are always QWEP.

## Abstract

By adapting an ultraproduct technique of Junge and Zeng, we prove that radial completely bounded multipliers on $q$-Gaussian algebras transfer to $q$-Araki-Woods algebras. As a consequence, we establish the $w^{\ast}$-complete metric approximation property for all $q$-Araki-Woods algebras. We apply the latter result to show that the canonical ultraweakly dense C$^\ast$-subalgebras of $q$-Araki-Woods algebras are always QWEP.