Let $V$ is complex vector space with nondegenerate scalar product. Does $f\colon V \otimes V \to V$, such that $(f(a \otimes b),c) = (a,f(b \otimes c))$ and $f \neq 0$, exist? It is obvious that such function doesn't exist when $\operatorname{dim} V = 1$, but what about other dimensions?
It is also interesting for me what will be in case when base field is $\mathbb{R}$.
I'm guessing that it's supposed to be clear from the context that you're actually asking about the existence of a linear $f$ satisfying that identity. In any case, if you add the condition that $f$ be linear then there is no such map: $(f(x\otimes y),z)$ is linear in $y$, while $(x,f(y\otimes z))$ is conjugate-linear in $y$.