Let $\mathcal A$ be a Type $II_1$ von Neumann algebra, equipped with a finite trace $\tau: \mathcal A \to \mathbb C$. Further, denote by $\mathcal A^\times \subset \mathcal A$ the group of invertible elements. It is well-known that $\tau$ induces a *positive-valued* determinant function $\det: \mathcal A^\times \to \mathbb R^+$, that is, among many other properties, multiplicative, i.e \begin{equation} \det(AB) = \det(A)\det(B). \end{equation} 
**Question: Does there exist a non-trivial, multiplicative determinant function
$\det: \mathcal A^\times \to \mathbb C$, such that $\det(\mathcal A^\times) \cap (\mathbb C \setminus \mathbb R_{\geq 0}) \neq \emptyset$. If not, what exactly goes wrong ?**