For a counter-example let $X$ be a ternary $C^*$-ring which is an operator space (e.g. any closed subspace $X$ of $B(H)$ such that $XX^*X\subseteq X$) and define a new ternary operation by $$\{x,y,z\}=-[x,y,z].$$ It is a fundamental result of Zettl [1] that trenary $C^*$-rings decompose uniquely as a "positive" part and a "negative" part.

[1] <cite authors="Zettl, Heinrich">_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).</cite>