Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M\_n(\mathbb{C})$ has no characters and also $B(H)$, where $H$ is a Hilbert space.



About the existence of characters on $B(X)$
I guess that you mean that $B(H)$ has no character (=continuous unital algebra homomorphism into $\mathbf{C}$) if $H$ has dimension $\neq 1$ (idem for $M\_n(\mathbf{C})$ for $n\neq 1$), and thus that your question assumes $\dim(X)\ge 2$ (and hence $=\infty$).


Argyros and Haydon (Acta Math, 2011: [arXiv](https://arxiv.org/abs/0903.3921), [Project Euclid unrestricted access](https://projecteuclid.org/euclid.acta/1485892527)) constructed a Banach space $X$ of infinite dimension in which every bounded self-operator is scalar+compact. Hence for such a space, modding out by the ideal of compact self-operators yields a character.


