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 ternary $C^*$-rings decompose uniquely as a "positive" part and a "negative" part.
It is interesting to remark that you do not change the structure of an algebra by inserting a minus sign in front of the product because 2 is even, but you do change a ternary $C^*$-ring by doing so because 3 is odd!
[1] Zettl, Heinrich, A characterization of ternary rings of operators, Adv. Math. 48, 117-143 (1983). ZBL0517.46049.
