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# PhD Thesis Defence

Tuesday, October 22, 2019 — 2:00 PM EDT

Zack Cramer, Department of Pure Mathematics, University of Waterloo

"Compressible Matrix Algebras and the Distance from Projections to Nilpotents"

In this talk 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. Such algebras are said to be \textit{projection compressible}.

Using a case-by-case analysis based on reduced block upper triangular forms, we provide a complete description of the unital idempotent compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ up to similarity and transposition. By studying the similarity orbits of the remaining non-idempotent compressible algebras, we find that our list also represents all unital projection compressible algebras that exist in this setting. A variation on our analysis in $\mathbb{M}_3(\mathbb{C})$ leads to similar findings for unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$. We therefore deduce 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.

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 derive an explicit formula for the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$. 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 $Q$. Furthermore, we prove 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.

MC 5479

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