# Banach spaces admitting no proper quasi-affinity

I am interested in examples of Banach spaces $X$ satisfying the following two conditions:

(1) Every (continuous linear) injective operator $T:X\to X$ with dense range is surjective.

(2) $X$ is isomorphic to its hyperplanes.

Observations:

(a) A space satisfying these two conditions must be non-separable [GK], and it cannot admit an infinite dimensional separable complemented subspace.

(b) There are non-separable H.I. spaces that satisfy (1), but H.I. spaces fail (2).

(b) In [AKo] we can find an example of a non-separable $C(K)$ space in which every injective operator is surjective, but it fails (2).

• This is related to the question mathoverflow.net/questions/101253, but we (Nasseri, Schechtman, Tkocz, and I) now know that $\ell_\infty$ fails (1) (that solves Problem 1 in your book with Martinez-Abejon). The paper should be ready soon; I'll send you a copy when it is done if we don't immediately post it on the ArXiv. – Bill Johnson Jun 13 '14 at 16:42
• So $X$ cannot contain copies of $\ell_\infty$. – M.González Jun 14 '14 at 9:48