A representation theorem for algebras with commuting involutions (Q1971033)

Given a field \(K\) with an involution\ \(\bar{\;}\)\ and an element \(\varepsilon\in K\) satisfying \(\varepsilon\bar\varepsilon=1\), an \(\varepsilon\)-Hermitian form on a \(K\)-vector space \(V\) is a map \(\langle\;,\;\rangle:V\times V\to K\) that is \(K\)-linear in the first variable and satisfies \(\langle v,v'\rangle=\varepsilon \overline{\langle v',v\rangle}\) for \(v,v'\in V\). Passing from a linear operator on \(V\) to its adjoint with respect to the form \(\langle v,v'\rangle\) is an involution on the algebra End\({}_{K}(V)\) of \(K\)-linear operators on \(V\) called the adjoint involution induced by \(\langle v,v'\rangle\). The following is the main result. Let \(A\) be a finite-dimensional \(K\)-algebra with two commuting involutions. Given \(\varepsilon,\varepsilon'\in K\) satisfying \(\varepsilon\bar\varepsilon= \varepsilon'\bar{\varepsilon'}=1\), there exists a finite-dimensional \(K\)-vector space \(W\), a linear operator \(F\in\operatorname {End}_{K}(W)\), and nondegenerate \(\varepsilon\)- and \(\varepsilon'\)-Hermitian forms on \(W\) such that \(A\) is isomorphic to the centralizer of the smallest \(K\)-subalgebra of End\({}_{K}(W)\) containing \(F\) and closed under the adjoint involutions induced by the two forms. The result and its proof are inspired by \textit{H.-G. Quebbemann} [Linear Algebra Appl. 94, 193-195 (1987; Zbl 0621.16013)]. The authors indicate how to extend their main result to finite-dimensional algebras with finitely many commuting involutions.