Laplace transform and the noncommutative version of the Paley-Wiener theorem (Q1603089)

Let \(G\) be a unimodular locally compact group of type I with Haar measure \(\mu\) and with dual space \(\widehat G\), let \(S\) be a generating subsemigroup with inner points, and let \(S_c=\{x\in G:\rho(x)\leq 1\) for all \(\rho\in S^*_l\}\), where \(S^*_l\) is the set of non-negative bounded semi-characters of the semigroup \(S\). A semi-character \(\rho:S\to[0,1]\) may be extended to a homomorphism \(\rho:G\to [0,\infty)\). If \(\rho_0S= \{p\lambda: \lambda \in\widehat G,\rho\in S^*_l,\rho\leq \rho_0\}\) and if \(f\in L^1(S,\rho_0 d\mu)\), then the Laplace transform \({\mathcal L}(f)\) is defined by \({\mathcal L}(f) (\rho\lambda)={\mathcal F}(f\rho)(\lambda)\), where \({\mathcal F}\) is the Fourier transform on \(L^1(G)\). In the main theorem of this paper, conditions are stated on \(S,G\), to ascertain that \({\mathcal L}\) is an isometric isomorphism between \(L^2(S_c)\) and \(H^2(\widetilde{S_0})\), where \(\widetilde S_0=\widetilde S\setminus\widehat G\) and \(\widetilde S\) is set of representations of the semigroup \(S\) in the form \(\rho\lambda\), \(\rho\in S^*_l\), \(\lambda\in\widehat G\).