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 $\mathcal{R}$ be a von Neumann algebra, and $\Phi$ be a normal conditional expectation from $\mathcal{R}$ onto a von Neumann subalgebra $\mathcal{S}$ of $\mathcal{R}$. Let $\mathfrak{R}[T]$ denote the range projection of an operator $T$. For a positive operator $A$ in $\mathcal{R}$, we prove that $\Phi (\mathfrak{R}[A])\le \mathfrak{R}[\Phi (A)]$ with equality if and only if $\mathfrak{R}[A]\in \mathcal{S}$.

5 pages. corrected hypothesis of Theorem 2.1 (Jensen' operator inequality) - `unital' completely positive map