Qualitative uncertainty principles for groups with finite dimensional irreducible representations (Q1028346)

Let \(G\) be a unimodular locally compact group equipped with a left Haar measure \(m_G\). Let \(\widehat{G}\) denote the dual space of \(G\), that is, the set of equivalence classes of irreducible unitary representations of \(G\), endowed with Fell's topology. Suppose that there exists a (necessarily uniqe) measure \(\mu_G\) on \(\widehat{G}\) such that the Plancherel formula holds. For \(f\in L^1(G)\), let \[ A_f=\{x\in G: f(x)\neq 0\} \quad\text{and}\quad B_f=\{\pi\in \widehat{G}:\pi(f)\neq 0\}. \] \(G\) is said to satisfy the qualitative uncertainty principle (QUP) if for every \(f\in L^1(G)\cap L^2(G)\), \(m_G(A_f)< m_G(G)\) and \(\mu_G(B_f)< \mu_G(\widehat{G})\) implies that \(f=0\) a.e.. In this paper the author proves that if \(G\) is a Moore group which is neither compact nor discrete, then \(G\) satisfies the QUP if and only if the connected component of the identity of \(G\) is noncompact.