Does there exists a $C^*$-algebra corresponding to every Banach ternary algebra?

Question

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.

Answer 1

This is far from an area I am an expert in, but I would keep in mind the following class of examples.

Let $A$ be any Banach $*$-algebra. So $A$ is a Banach algebra, and $A$ is a $*$-algebra, and the $*$-operation is continuous. Then we can renorm $A$ to make the $*$-operation isometric: $\|a^*\| = \|a\|$. For example, $A$ could be a $C^*$-algbera; or something well-behaved but non-$C^*$, like $L^1(G)$ for a locally compact group $G$; or something very degenerate, like a complex Banach space $E$ with an involution, given the zero product.

Turn $A$ into a ternary Banach algebra by defining the triple product as $[a,b,c] = ab^*c$. We now see that it's going to be hopeless to study $A$ using just $C^*$-algebras. For example, in the language of the paper you link to, in the highly degenerate case $A$ may not have any non-zero $C^*$-seminorms.

As Yemon says in the comments, in some cases it might be possible to build a "linking algebra" in the framework of Banach $*$-algebras: my guess is that maybe this would work for the $L^1(G)$ example. However, we should keep in mind that a general Banach algebra can have rather little structure, and the same will be true of ternary Banach algebras.