A note on the diagonal maximality of operator algebras

A subalgebra A of B(H) is said to be maximal with respect to its diagonal if it cannot be properly contained in any other subalgebra with the same diagonal. In this paper, we show that if T is a hyperreducible, maximal triangular algebra with a totally-atomic or nonatomic diagonal D, then, for a given separating vector ξ of D, the algebra of all operators T in T having ξ as an eigenvector is maximal respect to its diagonal. We also prove that each reflexive algebra defined by a double triangular lattice of projections in a matrix algebra has the diagonal maximality if the double triangular lattice of projections generates the whole matrix algebra.