About the existence of characters on $B(X)$ 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.
 A: Examples were known before the Argyros-Haydon space mentioned in Yves Cornulier's answer. For instance, if $J$ denotes the James space, then the image of the canonical map $J\to J^{**}$ has codimension $1$, and from this one can show that the closed 2-sided ideal $W(J)$ of all weakly compact operators on $J$ has codimension one in $B(J)$, thus giving us a character.
I am not sure where this was first observed: I learned of it from this paper of Loy and Willis:

MR1044280 (91f:46069)
  R. J. Loy, G. A Willis. Continuity of derivations on B(E) for certain Banach spaces E.
  J. London Math. Soc. (2) 40 (1989), no. 2, 327–346.

In the same paper, they exhibit some other examples of $E$ for which $B(E)$ admits a closed ideal of codimension $1$. For instance, one can take $K$ to be the scattered compact space obtained from the interval $[0,\omega_1]$ in the order topology, and then $E=C(K)$ has this property. (An alternative construction of the character for $B(C(K))$ can also be found in the 2014 paper of Kania, Koszmider and Lauststen, "A weak∗-topological dichotomy with applications in operator theory".) They also construct a James-type example of an $E$ such that $B(E)$ quotients on $\ell^\infty$, and hence has a lot of characters. All these examples are of a different flavour to the spaces that have been constructed more recently using the Argyros-Haydon machinery and its descendants.
At the other extreme, it is worth noting that if $X=\ell_p$ ($1\leq p\leq\infty$) or $X=L_p[0,1]$ ($1\leq p \leq\infty$) then $B(X)$ has no characters.
A: In order to complement answers given by Ycor and Yemon Choi, let me mention the space $X_M$ constructed by Mankiewicz (Isreael J. Math., 1989), which has the following remarkable properties:


*

*$X_M$ is separable and super-reflexive,

*$B(X_M)$ has a continuous homomorphism onto $\ell_\infty$.


Since $\ell_\infty$ has $2^{2^{\aleph_0}}$ characters, so has $B(X_M)$.
Another example of a space $X$ for which $B(X)$ has a character is G. Edgar's long James space. Recently, many spectacular separable examples of spaces whose algebras of operators admit characters have been constructed so the list goes on.
A: I think that this question is a suitable reason to remind everyone about a very interesting (but not so well known) unpublished paper C.J. Read, "Different forms of the approximation property", Unpublished manuscript, approximate date: 1989. In this paper Read shows the existence of Banach spaces with "approximate characters". The techniques used by Read is of the same type as the techniques of Mankiewicz in the paper mentioned in the answer of Tomek Kania. The techniques was introduced by Efim Gluskin in his papers in 1981.
(I posted Read's paper on my web page because one of my papers relies heavily on Read's construction, which is not available anywhere.)
A: 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, Project Euclid unrestricted access) 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.
