An operator inequality for range projections

Abstract

By a result of Lundquist-Barrett, it follows that the rank of a positive semi-definite matrix is less than or equal to the sum of the ranks of its principal diagonal submatrices when written in block form. In this article, we take a general operator algebraic approach which provides insight as to why the above rank inequality resembles the Hadamard-Fischer determinant inequality in form, with multiplication replaced by addition. It also helps in identifying the necessary and sufficient conditions under which equality holds. Let $\mathscr{R}$ be a von Neumann algebra, and $Φ$ be a normal conditional expectation from $\mathscr{R}$ onto a von Neumann subalgebra $\mathscr{S}$ of $\mathscr{R}$