This fact is an easy consequence of results of the paper by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpler proof?
In this particular case, you can proceed similarly as in Example 2.1 here. For more general results please consult a 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, 131–193. Contemp. Math. 85, 1989.