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 $\mathscr{B}(X)$.