Vol. 21, No. 3, 1967

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A theorem on random Fourier series on noncommutative groups

Alessandro Figà-Talamanca and Daniel Rider

Vol. 21 (1967), No. 3, 487–492

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 < .

Mathematical Subject Classification
Primary: 22.65
Secondary: 42.00
Received: 15 March 1966
Published: 1 June 1967
Alessandro Figà-Talamanca
Daniel Rider