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 $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$
I was informed that by $3.2$ Proposition of this paper and the fact that double dual of $C^*$-algebra is again $C^*$-algebra it follows that
$X$ is ternary $C^*$-ring then $X^{**}$ is also ternary $C^*$-ring. I'm unable to see it from the theorem. Can someone please explain me the logic behind this?
Thanks in advance.
Edit: $X$ is called associative provided $$[[a,b,c],d,e]=[a,[d,c,b],e]=[a,b, [c,d,e]]$$
Proposition $3.2$: Let $(X, [...],\|.\|)$ be a Ternary $C^*$-ring. Then there exists a unique pair $(A,a)$ satisfying
$(1)$ $A$ is a $C^*$-algebra and $X$ is a right Banach $A$-module.
$(2)$ $a: X \times X \to A$ is conjugate bilinear with $\|a\| \leq 1$ and $a(x.b, y)=a(x,y)b$ and $a(x,y)^*=a(y,x)$
$(3)$ $[x,y,z]=x.a(z,y)$
$(4)$ The linear hull of $a(X,X)$ is norm dense in $A$.