Complete mixing for convolution operators

Establish whether strict aperiodicity of a probability measure μ on a locally compact group G characterizes complete mixing of the convolution operator λ^1(μ) on L^1(G), i.e., determine if and under what conditions lim_{n→∞} ||λ^1(μ)^n f||_1 = 0 for every f ∈ L^1(G) is equivalent to μ being strictly aperiodic (its support not contained in any translate of a proper closed normal subgroup).

Background

The paper resolves the complete mixing problem for multipliers: Theorem 4.7 shows that for φ ∈ P(G), complete mixing of the sequence (Mφ^n) is equivalent to φ being strictly aperiodic. This parallels classical support-based characterizations in harmonic analysis.

In contrast, the analogous characterization for convolution operators associated to probability measures on locally compact groups remains unsettled. The authors point to this as an open direction, referencing their earlier work for further discussion.