Abstract :
Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X)
as an algebra over 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?
