Let G be a compact group.
For x ∈ G we shall consider a formal Fourier series (∗)∑diTr(UiAiDi(x)) where the
Di are distinct (non equivalent) irreducible representations of G of degree di, Ui are
arbitrary unitary operators and Ai fixed linear transformations on the Hilbert space
of dimension di and Tr denotes the ordinary trace. We shall prove that
∑diTr(AiAi∗) < ∞, provided that (∗) represents a function in L1(G) for all
U = {Ui} belonging to a set M which has positive Haar measure in the
group G=∏𝒰(di), where 𝒰(di) is the group of all unitary operators on the
di-dimensional space. If we think of G as a probability space, with respect to its
Haar measure, then (∗) is a Fourier series with “random coefficients” and the
result can be stated in the following way: if (*) represents, with positive
probability, a function in L1(G) then ∑diTr(AiAi∗) < ∞. An earlier result
of the authors implies then that, under the same hypothesis, (∗) is, with
probability one, the Fourier series of a function belonging to Lp(G) for every
p < ∞.
