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 $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$

Does there exist a unique $C^*$-algebra corresponding to each ternary $C^*$-ring?

$3.2$ Proposition of this paper seems to answer what i am looking for. The construction given in paper goes as follows: For $y,z \in X$, consider the bounded linear map $D_{y,z}: X \to X$ defined as $D_{y,z}(x)= [x,y,z]$. Let $V=$ span $\{D_{y,z}: y,z \in X \} \subset L(X)$. After defining like this, the author shows that $V$ is pre $ C^*$-algebra. Finally, author considers the opposite algebra of the norm closure of $V$ to construct the required $C^*$-algebra.

Can someone please explain me why do we need to consider opposite algebra and the motivation behind this proof?

key pointis the axiom $\Vert [x,x,x,] \Vert= \Vert x\Vert^3$ which is an analogue of the Cstar condition that is one of the axioms for a Cstar algebra. Calling these objects "ternary Banach algebras", as you have done, is misleading since that name should be reserved for an object which does not have the "3-variable Cstar condition" $\endgroup$3more comments