TITLE: Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
QUESTION [9 upvotes]: The space $( \ell^2 ,\lVert \cdot \rVert _2 )$  is a Hilbert space. The space 
 $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any closed convex bounded subset $C\subseteq X $ and any nonexpansive map $T:C\to C $ there is a $x\in C$ such that $T(x)=x$)
The space $X$ isn't uniformly convex so I can't use theorems about uniformly convex. There is no theorems about F.P.P in products of spaces and I have no another idea. Does someone have any idea?

REPLY [8 votes]: I think that the answer is yes, and that it should follow from the following facts:

every Hilbert space is uniformly convex, hence it has normal structure;
the direct sum of two Banach spaces with normal structure, endowed with the infinity norm, has again normal structure (Belluce-Kirk-Steiner, Pacific Journal Math. 1968);
the finite direct sum of separable Banach spaces (with any of the possible equivalent norm on it) is itself reflexive, in particular $X = \ell^2 \oplus \ell^2$ is reflexive; 
normal structure on a reflexive Banach space implies FPP (Kirk,  Amer. Math. Monthly 1965).