# Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?

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?

• It's an obvious consequence, modulo Zettle's theorem, of the same question for C*-algebras. Sep 7 at 8:58
• @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? Sep 7 at 10:10
• You should actually read Zettle's paper. The answer to your second question is no and you can read it from Zettle's paper Sep 8 at 9:03