Let $V$ be a TRO (closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V)$, $D(V)$ denote the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(V)$, the linking $C^*$-algebra of $V$ as follows:

$$A(V) = \begin{bmatrix} C(V) & V\\ V^* & D(V) \end{bmatrix}$$ Using this linking $C^*$-algebra one obtains a functor from category of TROs to the category of $C^*$-algebras using which one studies representation theory, nuclearity, exactness and ideal theory of TROs. Moreover, One can show that two TROs are isomorphic if and only if their corresponding linking $C^*$-algebras are isomorphic.

Recall that a ternary Banach algebra is a complex associative Banach space $A$, equipped with a ternary product $[.,.,.]:A^3 \to A$ which is linear in outer variables and conjugate linear in middle variable and $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$ Since TROs are obvious examples of ternary Banach algebra. This motivates me to ask following:

Does there exist $C^*$-algebra corresponding to each ternary Banach algebra "similar" to the one we have for TROs?

P.S: Above mentioned question might be some known result. Since I have started looking at ternary Banach algebra recently only so I have not seen such result anywhere.

Let $V$ be a TRO (closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V)$, $D(V)$ denote the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(V)$, the linking $C^*$-algebra of $V$ as follows:

$$A(V) = \begin{bmatrix} C(V) & V\\ V^* & D(V) \end{bmatrix}$$ Using this linking $C^*$-algebra one obtains a functor from category of TROs to the category of $C^*$-algebras using which one studies representation theory, nuclearity, exactness and ideal theory of TROs. Moreover, One can show that two TROs are isomorphic if and only if their corresponding linking $C^*$-algebras are isomorphic.

Recall that a ternary Banach algebra is a complex associative Banach space $A$, equipped with a ternary product $[.,.,.]:A^3 \to A$ which is linear in outer variables and conjugate linear in middle variable and $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$ Since TROs are obvious examples of ternary Banach algebra. This motivates me to ask following:

Does there exist $C^*$-algebra corresponding to each ternary Banach algebra "similar" to the one we have for TROs?

Edit: This paper (section $3$) discusses what i'm looking for but unfortunately I don't get any intuition for the given construction of $C^*$-algebra. I would be glad if someone can explain me the construction given in the paper.

Let $V$ be a TRO( closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V), D(V)$ denotes the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(V)$, the linking $C^*-$ algebra of $V$ as follows:

$$A(V) = \begin{bmatrix} C(V) & V\\ V^* & D(V) \end{bmatrix}$$ Using this linking $C^*-$ algebra one obtains a functor from category of TROs to the category of $C^*-$algebras using which one studies representation theory, nuclearity, exactness and ideal theory of TROs. Moreover, One can show that two TROs are isomorphic if and only if their corresponding linking $C^*-$algebras are isomorphic.

Recall that a ternary Banach algebra is a complex associative Banach space $A$, equipped with a ternary product $[.,.,.]:A^3 \to A$ which is linear in outer variables and conjugate linear in middle variable and $$\vert\vert[a,b,c]\vert \vert \leq \vert \vert a \vert \vert \vert \vert b\vert \vert \vert \vert c\vert \vert$$ Since TROs are obvious examples of ternary Banach algebra. This motivates me to ask following:

Does there exist $C^*-$algebra corresponding to each ternary Banach algebra "similar" to the one we have for TROs?

P.S: Above mentioned question might be some known result. Since I have started looking at ternary Banach algebra recently only so I have not seen such result anywhere.

Let $V$ be a TRO  (closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V)$, $D(V)$ denote the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(V)$, the linking $C^*$-algebra of $V$ as follows:

$$A(V) = \begin{bmatrix} C(V) & V\\ V^* & D(V) \end{bmatrix}$$ Using this linking $C^*$-algebra one obtains a functor from category of TROs to the category of $C^*$-algebras using which one studies representation theory, nuclearity, exactness and ideal theory of TROs. Moreover, One can show that two TROs are isomorphic if and only if their corresponding linking $C^*$-algebras are isomorphic.

Recall that a ternary Banach algebra is a complex associative Banach space $A$, equipped with a ternary product $[.,.,.]:A^3 \to A$ which is linear in outer variables and conjugate linear in middle variable and $$\|[a,b,c]\| \leq \| a \| \| b\|\| c\|$$ Since TROs are obvious examples of ternary Banach algebra. This motivates me to ask following:

Does there exist $C^*$-algebra corresponding to each ternary Banach algebra "similar" to the one we have for TROs?

P.S: Above mentioned question might be some known result. Since I have started looking at ternary Banach algebra recently only so I have not seen such result anywhere.