Order preserving and order preserving operators on the class of convex functions in Banach spaces

Recently S. Artstein-Avidan and V. Milman have developed an    
abstract duality theory and proved the following remarkable    
result: up to linear terms, the only fully order preserving    
operator (namely, an invertible operator whose inverse also    
preserves the pointwise order between functions) acting        
on the class of lower semicontinuous proper convex functions   
defined on R^n is the identity operator, and the only fully    
order reversing operator acting on the same set is the         
Fenchel conjugation (Legendre transform). We establish         
a suitable extension of their result to infinite dimensional   
Banach spaces.                                                 
                                                               
This is a joint work with Alfredo N. Iusem and Benar F. Svaiter