Bounded Operators on Banach Spaces : Kernels, Closed Ideals, and Uniqueness of Quotient Algebra Norms

Arnott, Max and Laustsen, Niels (2024) Bounded Operators on Banach Spaces : Kernels, Closed Ideals, and Uniqueness of Quotient Algebra Norms. PhD thesis, Lancaster University.

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Abstract

This thesis is comprised of four chapters. Chapter 1 consists of preliminary definitions and descriptions of the notation we will be using throughout. In Chapter 2, we ask the following question: `for a given Banach space $X$ and an arbitrary closed subspace $Y$ of $X$, is there necessarily an operator $T\in \mathscr{B}(X)$ for which $\ker T = Y$?' We prove that the answer to this question is yes when $X = c_0(\Gamma)$ or $X = \ell_p(\Gamma)$ for $\Gamma$ uncountable and $1 In Chapter 3, we classify the lattice of closed ideals of the space of bounded operators on the direct sums $X= \left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{c_0} \oplus c_0(\Gamma)$ and $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{\ell_1} \oplus \ell_1(\Gamma)$ for every uncountable cardinal $\Gamma$. In Chapter 4, we let $X$ be one of the following Banach spaces, for which we know the entire lattice of closed ideals of the Banach algebra $\mathscr{B}(X)$ of bounded operators on $X$: \begin{itemize} \item $X= (\ell_2^1\oplus \ell_2^2 \oplus\cdots\oplus \ell_2^n\oplus\cdots)_{c_0}$ or $X= (\ell_2^1 \oplus \ell_2^2 \oplus \cdots \oplus \ell_2^n \oplus \cdots )_{\ell_1}$, \item $X= (\ell_2^1 \oplus \ell_2^2 \oplus \cdots \oplus \ell_2^n \oplus \cdots)_{c_0}\oplus c_0(\Gamma)$ or $X=(\ell_2^1 \oplus \ell_2^2 \oplus \cdots \oplus \ell_2^n \oplus \cdots)_{\ell_1}\oplus\ell_1(\Gamma)$ for an uncountable index set~$\Gamma$, \item $X = C_0(K_{\mathcal{A}})$, the Banach space of continuous functions vanishing at infinity on the locally compact Mr\'{o}wka space~$K_{\mathcal{A}}$ associated with an almost disjoint family~$\mathcal{A}$ of infinite subsets of~$\mathbb{N}$, constructed such that $C_0(K_{\mathcal{A}})$ admits `few operators'. \end{itemize} We show that in each of these cases, the quotient algebra $\mathscr{B}(X)/\mathscr{I}$ has a unique algebra norm for every closed ideal $\mathscr{I}$ of $\mathscr{B}(X)$.