This fact is an easy consequence of results of the [paper](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-90/issue-2/Classes-of-Banach-spaces-with-unique-isometric-preduals/pjm/1102778987.full) by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpler proof?



Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces
In this particular case, you can proceed similarly as in Example 2.1 [here](https://web.archive.org/web/20100610042345/http://math.uchicago.edu/%7Eamwright/DavidsonWright2.pdf). For more general results please consult a fantastic survey on unique preduals by G. Godefroy:



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> 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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