Realizing irreducible semigroups and real algebras of compact operators (Q950464)

From the abstract: Let \({\mathcal B}(X)\) be the algebra of bounded operators on a complex Banach space \(X\). Viewing \({\mathcal B}(X)\) as an algebra over \(\mathbb R\), we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?