On the product of Riesz sets in dual objects of compact groups (Q2751529)

Let \(G\) be a compact group and \(\widehat G\) its dual object, i.e. the set of equivalence classes of all irreducible continuous unitary representations. For \(E\subseteq\widehat G\), \(M_E(G)\) denotes the set of all measures on \(G\) whose Fourier transform vanishes outside \(E\). In this paper, the set \(E\) is called \(s\)-small \(p\) set if for any \(\mu_1,\dots,\mu_p\in M_E(G)\) the convolution \(\mu_1*\dots*\mu_p\) is absolutely continuous with respect to Haar measure. An \(s\)-small 1 set is called a Riesz set due to the well-known Riesz theorem concerning measures on the circle. It is proved that the product \(E_1\times E_2\) of Riesz sets \(E_k\) for compact groups \(G_k\), \(k=1,2\), is the Riesz set for \(G_1\times G_2\). This is an analogue of the Bochner theorem for measures on the 2-torus. It is proved that \(E\subseteq\widehat G\) is an \(s\)-small 2 set if for any \(\sigma,\tau\in\widehat G\) the set \((\sigma\times E)\mathop\cap(\tau\times\overline E)\) is finite, where \(\times\) denotes the product and the bar denotes conjugation, and a similar result for each \(p\) is obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].