Représentations unitaires et opérateurs à trace. (Unitary representations and traceable operators)
The author proves the following generalization of Theorem 4.2.2 of \textit{N. V. Pedersen} [Math. Ann. 247, 191-244 (1980; Zbl 0406.22008)]. Theorem. Let G be a connected simply connected Lie group and let \({\mathfrak g}\) be its algebra. There exists a non-zero semi-invariant P in S(\({\mathfrak g}_{{\mathbb{C}}})\) (we denote by w the semi-invariant of U(\({\mathfrak g}_{{\mathbb{C}}})\) corresponding to P and by O the set \(\{\ell \in g^*_{{\mathbb{C}}}:\) \(P(-i\ell)=0\})\), such that: for every normal factorial representation \(\pi\) of G, for which there are f in O and \(\tau \in X_ G^{irr}(f)\) with \(\ker\pi= \ker T^ T_{f,\tau}\), we have: \(\pi(w*\phi)\) is traceable relative to the factor \(\pi(G)''\) generated by \(\pi(G)\), where \(\phi\) is an arbitrary element of \({\mathcal D}(G)\).
scientific article; zbMATH DE number 4055936
scientific article