On semigroups of matrices with traces in a subfield (Q1336422)

It is shown that an irreducible semigroup of \(n \times n\) complex matrices with real spectra is conjugate to a semigroup of real matrices. A weaker result is obtained for a semigroup of matrices over an arbitrary field with traces in a subfield. Reviewer's note: Lemma 1c) is valid only under the additional assumption that the \(K\)-algebra \(\text{Alg}_ K({\mathcal S})\) is finite dimensional. This does not affect the validity of the main results.