Preduals and $c_0$ I know that $c_0$ does not have a predual, but if we put an equivalent norm on $c_0$, can this space have a predual?

No.  In a separable dual space, every closed convex set is the closed convex hull of its extreme points.  (Krein-Milman property).  But in $c_0$ with an equivalent norm, the original unit ball is still a closed convex set with no extreme points.
Reference  
Diestel, J.; Uhl, J. J. jun., Vector measures, Mathematical Surveys. No. 15. Providence, R.I.: American Mathematical Society (AMS). XIII, 322 p. $ 35.60 (1977). ZBL0369.46039.
