dc.contributor.author Cramer, Zachary dc.date.accessioned 2019-11-15 13:38:43 (GMT) dc.date.available 2019-11-15 13:38:43 (GMT) dc.date.issued 2019-11-15 dc.date.submitted 2019-11-12 dc.identifier.uri http://hdl.handle.net/10012/15251 dc.description.abstract 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}. en 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. 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. dc.language.iso en en dc.publisher University of Waterloo en dc.subject matrix en dc.subject operator en dc.subject projection en dc.subject idempotent en dc.subject nilpotent en dc.subject compression en dc.subject projection compressible en dc.subject idempotent compressible en dc.title Compressible Matrix Algebras and the Distance from Projections to Nilpotents en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department Pure Mathematics en uws-etd.degree.discipline Pure Mathematics en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws.contributor.advisor Marcoux, Laurent uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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