Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following differential equation on $A$:
$$Z'=Z^2-Z.\tag{*}\label{star1}$$
(For $A=M_n(\mathbb{C})$ one can easily check that $$D'=D(T-n)\tag{**}\label{star2}$$ where $D$, $T$ are the standard determinant and trace respectively and $D'$ is the derivative of $D$ along solutions of \eqref{star1}. Note that $n$ in \eqref{star2} can be regarded as $\operatorname{trace}(I_n)$. We will modify this $n$ to $1$ in the case of normalized trace.
In fact "determinant" is the unique analytic function on $M_n(\mathbb{C})$ satisfying the equation \eqref{star2} with initial condition $D(I_n)=1$.)
We try to generalize this situation of matrix algebra to a $C^*$-algebra $A$ with a faithful normalized trace $T$. So we consider the following modified differential equation:$$D'=D(T-1)\tag{***}\label{star3}$$ where the unknown $D$ is a function on $A$ and $T$ is a normalized trace. Moreover $D'$ is the derivative of $D$ along solution of \eqref{star1}.
What can be said about existence of a global solution $D$ for \eqref{star3} with initial condition $D(1)=1$? Does such a solution $D$ satisfy the multiplicativity condition $D(ab)=D(a)D(b)$? Is $D^{-1}(0)$ equal to the set of all non invertible elements? As a motivation for the later question, we note that the group of invertible elements is flow-invariant under system \eqref{star1} (see On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra).
If there are no some complete answers to the above questions for an arbitrary algebra with a faithful normalized trace, what would be the answer of those questions in the particular case $A=C^*_{\text{red}} (G)$? For which kind of groups the answer to the above questions are known?
Remark: We conclude that "determinant" as a function on matrix algebra can be dynamically and uniquely extracted from "trace", at least in a neighborhood of the identity matrix since the identity matrix, as a singularity of \eqref{star1}, attracts all nearby orbits, as time goes to $-\infty$. This is somewhat compatible with the classical fact that "determinant" of a matrix $B$, as an invariant polynomial, can be generated by "trace" of powers of $B$, that is $\operatorname{trace}(B^k),\;k\in \mathbb{N}$. But this dynamical interpretation we provided, needs merely trace of power $1$ not higher powers. More precisely, if we denote by $\phi$ the flow of $(*)$, then for $B$ sufficiently close to identity matrix we have $$\operatorname{Det}(B)=\exp\left(\int_{-\infty}^0 (n-\operatorname{trace})(\phi_t(B))dt\right).$$ So knowing "trace" leads us to knowing "determinant".
