Completely bounded lacunary sets for compact non-abelian groups
Volume 230 / 2015
Studia Mathematica 230 (2015), 265-279
MSC: Primary 43A46; Secondary 46L07, 47L25.
DOI: 10.4064/sm8391-1-2016
Published online: 27 January 2016
Abstract
In this paper, we introduce and study the notion of completely bounded $\varLambda _{p}$ sets ($\varLambda _{p}^{\rm cb}$ for short) for compact, non-abelian groups $G$. We characterize $\varLambda _{p}^{\rm cb}$ sets in terms of completely bounded $L^{p}(G)$ 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 $\varLambda _{p} $ sets for all $p \lt \infty $, but are not $\varLambda _{p}^{\rm cb}$ for any $p\geq 4$. This is done by showing that the space of completely bounded $L^{p}(G)$ multipliers is a proper subset of the space of $L^{p}(G)$ multipliers.
