Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

Abstract

We show that for each of the following Banach spaces~$X$, the quotient algebra $\mathcal{B}(X)/\mathcal{I}$ has a unique algebra norm for every closed ideal $\mathcal{I}$ of $\mathcal{B}(X):$ - $X=({⨁}_{n\in \mathbb{N}}{\ell }_2^n{)}_{c_0}$\quad and its dual,\quad $X=({⨁}_{n\in \mathbb{N}}{\ell }_2^n{)}_{{\ell }_1}$, - $X=({⨁}_{n\in \mathbb{N}}{\ell }_2^n{)}_{c_0}\oplus c_0(\Gamma )$\quad and its dual, \quad $X=({⨁}_{n\in \mathbb{N}}{\ell }_2^n{)}_{{\ell }_1}\oplus {\ell }_1(\Gamma )$,\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 Mrówka 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~$\mathcal{B}(X)$ 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 $\mathcal{B}(X)\setminus \mathcal{I}$ with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.