Let G be a locally compact group and μ be a probability measure on G.
We consider the convolution operator λ1(μ):L1(G)→L1(G)
given by λ1(μ)f=μ∗f and its restriction λ10(μ) to
the augmentation ideal L10(G). Say that μ is uniformly ergodic if the
Ces\`aro means of the operator λ10(μ) converge uniformly to 0, that
is, if λ10(μ) is a uniformly mean ergodic operator with limit 0 and
that μ is uniformly completely mixing if the powers of the operator
λ10(μ) converge uniformly to 0.
We completely characterize the uniform mean ergodicity of the operator
λ1(μ) and the uniform convergence of its powers and see that there
is no difference between λ1(μ) and λ10(μ) in this
regard. We prove in particular that μ is uniformly ergodic if and only if
G is compact, μ is adapted (its support is not contained in a proper
closed subgroup of G) and 1 is an isolated point of the spectrum of μ.
The last of these three conditions is actually equivalent to μ being
spread-out (some convolution power of μ is not singular). The measure μ
is uniformly completely mixing if and only if G is compact, μ is
spread-out and the only unimodular value of the spectrum of μ is 1.Comment: Updated version. References to previous related results are improved.
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