In this particular case, you can proceed similarly as in Example 2.1 [here][1]. For more general results please consult fantastic survey on unique preduals by G. Godefroy:

> G. Godefroy. Existence and uniqueness of isometric preduals: a survey. In: BorLuh Lin, editor, Banach Space Theory. Proc. of the Iowa Workshop on Banach Space Theory 1987, pages 131–193. *Contemp. Math*. **85**, 1989.


  [1]: http://math.uchicago.edu/~amwright/DavidsonWright2.pdf