A ternary $C^{\ast}$-ring is a complex Banach space $X$, equipped with a ternary product $[\cdot,\cdot,\cdot]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle variable. Also $X$ is associative i.e. $$[[a,b,c],d,e]=[a,[d,c,b],e]=[a,b, [c,d,e]].$$ Moreover, $\lVert[a,a,a]\rVert= \lVert a\rVert^3$ and $\lVert[a,b,c]\rVert \leq \lVert a \rVert \lVert b\rVert\lVert c\rVert$.



> 
> Does there exist a ternary $C^{\ast}$-ring which is not an operator space?
> 
> 
> 


One obvious class of examples of ternary $C^{\ast}$-rings is the class of ternary rings of operators but they all are operator spaces.



Example of a ternary $C^{\ast}$-ring which is not an operator space
According to Zettl [1], a *ternary ring of operators* (TRO) is a
ternary $C^\*$-ring which is isomorphic to a closed
subspace $X\subseteq B(H)$, such that $XX^\*X\subseteq X$, equipped with the ternary multiplication
$$
 [x,y,z] := xy^\*z.
 $$
On the other hand, an *anti-TRO* is a ternary $C^\*$-ring defined as above, except that the multiplication operation is
$$
 [x,y,z] := -xy^\*z.
 $$
It is a fundamental result of Zettl [1] that every ternary $C^\*$-ring $X$ decomposes uniquely as
$$
 X=X\_+\oplus X\_-,
 $$
where $X\_+$ is a TRO, and $X\_-$ is an anti-TRO .


It seems to me that the reading of the question posed by the OP that makes the most sense is by taking the expression
"operator space" to mean a TRO. In this case the answer is yes, there does exist a ternary $C^\*$-ring which is not a
TRO: just take any non-zero anti-TRO. For an even more concrete example, take $X=M\_{n\times m}({\bf C})$, with ternary
multiplication $[x,y,z] := -xy^\*z$.


On the other hand, if one takes the expression "operator space" for its face value, Zettl's result implies that every
ternary $C^\*$-ring is an operator space in an even more canonical form than suggested by user @YemonChoi: write
$X=X\_+\oplus X\_-$, embedd $X\_+$ in $B(H\_+)$, and $X\_-$ in $B(H\_-)$, whence
$$
 X\subseteq B(H\_-\oplus H\_+).
 $$
This embeding preserves the operator space structure (norms on matrix algebras) that a TRO canonical possesses.


It is interesting to remark that if you change the (binary) multiplication operation on a $C^\*$-algebra by
$$
 x\circ y := -xy,
 $$
then the resulting object is strictly speaking a new C\*-algebra, but it is isomorphic to the old one. The isomorphism
is simply $a\mapsto -a$.


However, if you change the (ternary) multiplication on a ternary $C^\*$-ring by inserting a minus sign as above, then the
map $a\mapsto -a$ is no longer an isomorphism, essentially because 2 is even and 3 is odd!
Indeed, Zettl's uniqueness result tells you that the new ternary $C^\*$-ring
might not be isomorphic to the old one at all!




---


EDIT: Here are some details of Zettl's proof which might shed some light into the reason an anti-TRO not isomorphic to a TRO.


Given a ternary $C^\*$-ring $X$, let $A$ be the closed linear span within $B(X,X)$ (bounded operators on $X$) of the set
of operators of the form
$$
 T\_{y, z}:x\in X\mapsto [x,y,z]\in X,
 $$
as $y$ and $z$ range in $X$. It is easy to see that $A$ is a Banach algebra, and Zettl proves that $A$ is indeed a
$C^\*$-algebra for a unique involution operation "$^\*$" satisfying
$$
 T\_{y, z}^\* = T\_{z, y}.
 $$


Given this, it is clear that an operator of the form $T\_{y,y}$ is self-adjoint but the key question is whether or not
this is moreover positive.


If $X$ is a TRO, then $T\_{y, y}\geq 0$, while in the anti-TRO case, one has that $T\_{y, y}\leq 0$.


In other words, the positivity of $T\_{y, y}$ is a signature of TRO's not shared by their anti-TRO cousins.


[1] *Zettl, Heinrich*, [**A characterization of ternary rings of
operators**](http://dx.doi.org/10.1016/0001-8708(83)90083-X), Adv. Math. 48, 117-143
(1983). [ZBL0517.46049](https://zbmath.org/?q=an:0517.46049).


