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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?

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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.

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  • $\begingroup$ do you have electronic copy of this survey. Neither I, nor my institute don't have access to Contemporary Mathematics. $\endgroup$ – Norbert Jan 31 '14 at 12:58
  • $\begingroup$ Norbert, the whole paper is fully available (at least to me) through google books: books.google.co.uk/… $\endgroup$ – Tomasz Kania Jan 31 '14 at 18:23
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    $\begingroup$ unfortunately it is not fully avaliable for me even via your link $\endgroup$ – Norbert Jan 31 '14 at 20:40

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