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

TL;DR

This paper proves the uniqueness of algebra norms on quotients of bounded operator algebras for specific Banach spaces, showing all homomorphisms are continuous and have closed range, with implications for operator theory.

Contribution

It establishes the uniqueness of algebra norms on quotients of operator algebras for new classes of Banach spaces, extending previous results.

Findings

Every homomorphism from (X) is continuous and has closed range.

The identity operator factors through every non-ideal operator with controlled norms.

Quantitative factorization results are developed that may be of independent interest.

Abstract

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.