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Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle variable, $\|[a,a,a]\|= \|a\|^3$ and $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$

Is it known that second dual $X^{**}$ of ternary $C^*$-ring is again ternary $C^*$-ring?

Any references?

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    $\begingroup$ It's an obvious consequence, modulo Zettle's theorem, of the same question for C*-algebras. $\endgroup$ Sep 7 at 8:58
  • $\begingroup$ @NarutakaOZAWA: Would it be enough to observe that $A^{**}$ would be unique $C^*$-algebra corresponding to $X^{**}$ where $A$ is unique $C^*$-algebra obtained for $X$ by Zettle’s theorem. If yes, why so? $\endgroup$
    – Math Lover
    Sep 7 at 10:10
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    $\begingroup$ You should actually read Zettle's paper. The answer to your second question is no and you can read it from Zettle's paper $\endgroup$ Sep 8 at 9:03

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