TITLE: About the existence of characters on $B(X)$
QUESTION [13 upvotes]: 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.

REPLY [3 votes]: I think that this question is a suitable reason to remind everyone about a very interesting (but not so well known) unpublished paper C.J. Read, "Different forms of the approximation property", Unpublished manuscript, approximate date: 1989.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.)