Trace-preserving homomorphisms of semigroups.
The authors study the structure of trace-preserving homomorphisms of semigroups of trace-class operators. Let \({\mathcal S}\) be a semigroup of trace-class operators acting on a complex Hilbert space \(H\). If the closed subalgebra generated by \({\mathcal S}\) in the Banach algebra \({\mathcal C}_1\) of all trace-class operators is semisimple, then the semigroup \({\mathcal S}\) is called semisimple. The main result of the paper reads as follows. Let \(\phi: {\mathcal S} \to {\mathcal T}\) be a surjective trace-preserving homomorphism between semisimple semigroups of trace-class operators. If \({\mathcal S}\) and \({\mathcal T}\) satisfy certain totality conditions, then \(\phi\) is induced by a closed, injective operator \(V\) with dense domain and dense range in the following sense. For every \(S\in {\mathcal S}\), we have \(\phi(S)V=VS\) on the domain of \(V\). Some discussions and examples are given to understand the assumptions in this and in some preliminary results. Several applications of the main result in the finite and infinite dimensions are also presented. As for finite dimension, it is shown that every irreducible semigroup of operators individually similar to \(J\)-unitaries is similar to a semigroup of \(J\)-unitaries with one and the same \(J\). An analogous result holds for unitaries. As for infinite dimension, it is proved that if the elements of an irreducible semigroup of operators of the form \(F+\lambda I\), \(F\) being of finite rank, are individually similar to unitaries, then the whole semigroup is similar to a semigroup of unitaries via some densely defined, injective, positive operator with dense range.
scientific article; zbMATH DE number 2037115
scientific article