A representation theorem for algebras with involution (Q1090393)

By means of a short and clever argument, the author proves that any finite dimensional algebra with involution is the centralizer in a ring of vector space endomorphisms. Specifically, let A be a finite dimensional algebra over the field K. If A has an identity element and involution \({}^*\), then for any \(\epsilon\in K\) satisfying \(\epsilon \epsilon^*=1\), there is a finite dimensional vector space V and nondegenerate \(\epsilon\)-hermitian form h on V so that A embeds in \(R=Hom_ K(V,V)\), the adjoint mapping with respect to h on R extends \({}^*\), and for some \(f\in R\), \(A=C_ R\{f,f^*\}\), the centralizer in R of \(\{f,f^*\}\).