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.
Categories: math.FA