Compressible Matrix Algebras and the Distance from Projections to Nilpotents
Zachary Cramer · UWSpace (University of Waterloo) · 2019
In this thesis we address two problems from the fields of operator algebras and operator theory. In our first problem, we seek to obtain a description of the unital subalgebras $\\mathcal{A}$ of $\\mathbb{M}_n(\\mathbb{C})$ with the property that $E\\mathcal{A}E$ is an algebra for all idempotents $E\\in\\mathbb{M}_n(\\mathbb{C})$. Algebras with this property are said to be \\textit{idempotent compressible}. Likewise, we wish to determine which unital subalgebras of $\\mathbb{M}_n(\\mathbb{C})$ satisfy the analogous property for projections (i.e., self-adjoint idempotents). Such algebras are said to be \\textit{projection compressible}. We begin by constructing various examples of idempotent compressible subalgebras of $\\mathbb{M}_n(\\mathbb{C})$ for each integer $n\\geq 3$. Using a case-by-case analysis based on reduced block upper triangular forms, we prove that our list includes all unital projection compressible subalgebras of $\\mathbb{M}_3(\\mathbb{C})$ up to similarity and transposition. A similar examination indicates that the same phenomenon occurs in the case of unital subalgebras of $\\mathbb{M}_n(\\mathbb{C})$, $n\\geq 4$. We therefore demonstrate that the notions of projection compressibility and idempotent compressibility coincide for unital subalgebras of $\\mathbb{M}_n(\\mathbb{C})$, and obtain a complete classification of the unital algebras admitting these properties up to similarity and transposition. \t In our second problem, we address the question of computing the distance from a non-zero projection to the set of nilpotent operators acting on $\\mathbb{C}^n$. Building on MacDonald's results in the rank-one case, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\\mathbb{M}_n(\\mathbb{C})$ is $\\frac{1}{2}\\sec\\left(\\frac{\\pi}{\\frac{n}{n-1}+2}\\right)$. For each $n\\geq 2$, we construct examples of pairs $(Q,T)$ where $Q$ is a projection of rank $n-1$ and $T\\in\\mathbb{M}_n(\\mathbb{C})$ is a nilpotent of minimal distance to $Q$. Moreover, it is shown that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.