### Completely bounded lacunary sets for compact non-abelian groups

In this paper, we introduce and study the notion of completely bounded ${\Lambda}_{p}$ sets (${\Lambda}_{p}^{cb}$ for short) for compact, non-abelian groups G. We characterize ${\Lambda}_{p}^{cb}$ sets in terms of completely bounded ${L}^{p}\left(G\right)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are ${\Lambda}_{p}$ sets for all p < ∞, but are not ${\Lambda}_{p}^{cb}$ for any p ≥ 4. This is done by showing that the space of completely bounded ${L}^{p}\left(G\right)$ multipliers...