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?

$\begingroup$ For this particular space you can play the same game as in Example 2.1 here: math.uchicago.edu/~amwright/DavidsonWright2.pdf $\endgroup$– Tomasz KaniaAug 9 '13 at 11:20

$\begingroup$ @TomekKania I think you can post this as answer $\endgroup$– NorbertAug 9 '13 at 16:15
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.

$\begingroup$ do you have electronic copy of this survey. Neither I, nor my institute don't have access to Contemporary Mathematics. $\endgroup$– NorbertJan 31 '14 at 12:58

$\begingroup$ Norbert, the whole paper is fully available (at least to me) through google books: books.google.co.uk/… $\endgroup$ Jan 31 '14 at 18:23

1$\begingroup$ unfortunately it is not fully avaliable for me even via your link $\endgroup$– NorbertJan 31 '14 at 20:40