Trace-class unitary representation

A strongly continuous unitary representation $\pi$ of a finite-dimensional Lie group $G$ on $H$ is trace class if every integrated operator

\pi(f)=\int_G f(g)\pi(g)\,dg, \qquad f\in C_c^\infty(G),

is a trace-class operator. The integral uses a fixed left Haar measure.

What is required to have finite trace

The condition concerns the averages $\pi(f)$ . In an infinite-dimensional representation the individual unitaries $\pi(g)$ are not trace class: their absolute values equal the identity. Smooth averaging can nevertheless produce summable singular values.

Examples and a nonexample

Every finite-dimensional unitary representation is trace class. An infinite-dimensional trivial representation is not: choosing $f$ of integral one gives $\pi(f)=I$ . Thus a condition on the group alone does not force every one of its unitary representations to be trace class.

For finite-dimensional Lie groups, trace class is equivalent to nuclearity of the smooth-vector space.

References

Gerrit van Dijk, Karl-Hermann Neeb, Hadi Salmasian, and Christoph Zellner, On the characterization of trace class representations and Schwartz operators. §1, definition before Lemma 1.1 and Theorem 1.3.