We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case duals discrete groups under name “free products pairs” previous work ours. give several equivalent descriptions use them to establish properties like residual finiteness, Haagerup property or smash decomposition.

We study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., ...