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 $Tin mathscr{B}(X)$ 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 $X= left(bigoplus_{n in mathbb{N}} ell_2^nright)_{c_0} oplus c_0(Gamma)$ and $left(bigoplus_{n in mathbb{N}} ell_2^nright)_{ell_1} oplus ell_1(Gamma)$ 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 $X= (ell_2^1oplus ell_2^2 opluscdotsoplus ell_2^nopluscdots)_{c_0}$ 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 $X=(ell_2^1 oplus ell_2^2 oplus cdots oplus ell_2^n oplus cdots)_{ell_1}oplusell_1(Gamma)$ 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 $mathscr{B}(X)$.