Dixmier traces and unitary representations (Q5944770)

Let \(H_{1}, H_{2}\) be separable Hilbert spaces and let \(L(H_{1},H_{2})\) be the space of all bounded linear mappings from \(H_{1}\) to \(H_{2}\). \(J_{2}(H_{1},H_{2})\) denotes the subspace of \(L(H_{1},H_{2})\) of Hilbert-Schmidt mappings. Let \(G\) be a topological group and \(T_{1}, T_{2}\) be unitary representations of \(G\) into \(H_{1}\) and \(H_{2}\) respectively. The map \(A\mapsto T_{2}(g)A T_{1}(g^{-1})\) defines a unitary representation of \(G\). The space \(J_{2}(H_{1},H_{2})\) is a Hilbert space under the inner product \((A,B)= Tr(B^{*}A)\). This space is invariant under the representation described above.The sub-representation restricted to this Hilbert space is well known. In this paper the author describes a modification of this procedure and uses the Dixmier trace in place of the usual trace. The author studies this new representation for the group of second order real unimodular matrices in detail.