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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?

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  • $\begingroup$ The title could perhaps be a bit more explicit: It suggests to me more a general question on how to read a difficult proof, a question of Shakespearean nature in my opinion. $\endgroup$
    – Roland Bacher
    Commented Sep 1, 2021 at 7:10
  • $\begingroup$ @RolandBacher: Edited, thanks! $\endgroup$
    – Math Lover
    Commented Sep 1, 2021 at 7:18
  • $\begingroup$ You can also complete $V$ and obtain a $C^*$-algebra. The opposite $C^*$-algebra of this $C^*$-algebra is then simply the $C^*$-algebra constructed in the paper. $\endgroup$
    – user160032
    Commented Sep 1, 2021 at 8:44
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    $\begingroup$ Your terminology is not correct, if you look closely at the Zettl paper that you link to. What you call a "ternary Banach algebra" is what Zettl calls a "ternary Cstar ring", and the key point is 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" $\endgroup$
    – Yemon Choi
    Commented Sep 2, 2021 at 2:29
  • $\begingroup$ It would be good if you could fix the wording of your title and your question. Note also that you have written that X is a "complex associative Banach space" which does not make sense. $\endgroup$
    – Yemon Choi
    Commented Sep 2, 2021 at 2:30

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Yes, Proposition 3.2 from the article [MR0700979] that you link to gives you the answer for ternary $\rm C^*$-rings. The reason for taking the opposite algebra is that the algebra $V$ acts on the right of $X$ rather than on the left. A right action is the same thing as a left action by the opposite algebra. You can see this right action directly in (3) of this proposition.

As a special case, suppose $X$ is a right Hilbert $\rm C^*$-module over a $\rm C^*$-algebra $A$. Denoting the $A$-valued inner product on $X$ by $\langle x|y\rangle_A$ (which is linear on the right), we get a ternary $\rm C^*$-ring by $[x,y,z] := x\langle y|z\rangle_A$. In this setting, $D_{y,z}$ is right multiplication by $\langle y|z\rangle_A$. All ternary $\rm C^*$-rings are essentially like this one, except that the $A$-valued form need not be positive. (See the intro of the cited article, bottom of p118 after the displayed equation.)

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