Trace class groups (Q2810931)

Given a locally compact group \(G\), let \(C^\infty_c(G)\) be the locally convex space of test functions on~\(G\) (as introduced by \textit{F. Bruhat} [Bull. Soc. Math. Fr. 89, 43--75 (1961; Zbl 0128.35701)]), and \(dg\) be a Haar measure on~\(G\). An irreducible continuous unitary representation \(\pi\) of \(G\) is called trace class if \(\pi(f)=\int_G f(g)\pi(g)\,dg\) is a trace class operator for each \(f\in C^\infty_c(G)\). If every irreducible continuous unitary representation of \(G\) is trace class, then \(G\) is called a trace class group. For example, every compact group is trace class, and so is every locally compact abelian group. It is known from the work of \textit{A. A. Kirillov} that all connected nilpotent Lie groups are trace class [Russ. Math. Surv. 17, No. 4, 53--104 (1962; Zbl 0106.25001); translation from Usp. Mat. Nauk 17, No. 4(106), 57--110 (1962)], and \textit{Harish-Chandra} showed that every semisimple Lie group with finite centre and finitely many connected components is trace class [Trans. Am. Math. Soc. 76, 26--65, 234--253 (1954; Zbl 0055.34002)]. The authors gain many new insights into trace class groups. We mention sample results: Every trace class group is of type I (Theorem 1.7), but there exists a connected unimodular Lie group of type I which is not trace class (Proposition 1.9). All connected Lie groups with reductive Lie algebra are trace class (Theorem 2.1). A discrete group is trace class if and only if it is a Moore group (i.e., all of its irreducible continuous unitary representations are finite-dimensional), if and only if it is of type I, if and only if it is virtually abelian (Theorem 3.3). The direct product of two trace class groups is trace class (Proposition 1.10 (a)); some information on semi-direct products is also available.