  We show that for each of the following Banach spaces~$X$, the quotient
algebra $\mathscr{B}(X)/\mathscr{I}$ has a unique algebra norm for every closed
ideal $\mathscr{I}$ of $\mathscr{B}(X)\colon$
  - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}$\quad and its dual,\quad
$X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}$,
  - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}\oplus c_0(\Gamma)$\quad
and its dual, \quad $X=
\bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}\oplus\ell_1(\Gamma)$,\quad for
an uncountable cardinal number~$\Gamma$,
  - $X = C_0(K_{\mathcal{A}})$, the Banach space of continuous functions
vanishing at infinity on the locally compact Mr\'{o}wka space~$K_{\mathcal{A}}$
induced by an uncountable, almost disjoint family~$\mathcal{A}$ of infinite
subsets of~$\mathbb{N}$, constructed such that $C_0(K_{\mathcal{A}})$ admits
"few operators".
  Equivalently, this result states that every homomorphism
from~$\mathscr{B}(X)$ into a Banach algebra is continuous and has closed range.
The key step in our proof is to show that the identity operator on a suitably
chosen Banach space factors through every operator in
$\mathscr{B}(X)\setminus\mathscr{I}$ with control over the norms of the
operators used in the factorization. These quantitative factorization results
may be of independent interest.
