On weak containment properties

Abstract:

We prove, that two concepts of weak containment do not coincide, contradicting results in [1, Lemma 3.3 and Proposition 3.4]. The statement of Theorem 3.5 remains valid. There exist infinite tall compact groups $G$ (i.e. the set $\{ \sigma \in \hat G,\dim \sigma = n\}$ is finite for each positive integer $n$) having the mean-zero weak containment property. Such groups do not have the dual Bohr approximation property or $AP(\hat G) \ne C_\delta ^*(G)$.