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 $\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$.
$3.2$ Proposition of this paper assigns a $C^*$-algebra corresponding to ternary $C^*$-ring. 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 with involution $D_{y,z}^*=D_{z,y}$ Finally, author considers the opposite algebra of the norm closure of $V$ to construct the required $C^*$-algebra. We denote this $C^*$-algebra by $A(X)$.
Let $X$ and $Y$ be two ternary $C^*$-rings and $\pi: X \to Y$ be a ternary morphism. Given $\pi$ we get a $*$-morphism $A(\pi): A(X) \to A(Y)$. Is something known about the converse?
Under what conditions, every $*$-morphism from $A(X)$ to $A(Y)$ is of the form $A(\pi)$? What happens if $Y=B(H,K)$?
Any references or ideas?
