Reflexive algebras of matrices (Q1066232)

Let V be a finite dimensional complex vector space, \({\mathcal L}^ a \)lattice of subspaces of V, and \({\mathcal A}^ a \)subalgebra of Hom(V). Alg \({\mathcal L}\) is the algebra of all \(A\in Hom(V)\) which leave invariant every subspace \(W\in {\mathcal L}\) and Lat \({\mathcal A}\) is the lattice of all subspaces of V which are left invariant by every \(A\in {\mathcal A}\). The purpose of this paper is to study the conditions for \({\mathcal A}\) to be reflexive, i.e., Alg Lat \({\mathcal A}={\mathcal A}\). A semisimple subalgebra of Hom(V) is reflexive. Let \({\mathcal A}\) be a commutative algebra of \(n\times n\) matrices \((n=2,3)\) such that each \(A\in {\mathcal A}\) is reflexive, then \({\mathcal A}\) is reflexive \((n=2)\) and \({\mathcal A}\) is reflexive if and only if all \(A\in {\mathcal A}\) can be simultaneously diagonalized or dim \({\mathcal A}=2\) \((n=3)\). If \({\mathcal L}\) is a complemented lattice of subspaces of V, then Alg \({\mathcal L}\) is semisimple, which implies an algebraic proof of a result on reflexive subspace lattices due to \textit{K. J. Harrison} and \textit{W. E. Longstaff} [Indiana Univ. Math. J. 26, 1019-1025 (1977; Zbl 0387.46017)], where they used continuous geometries.