Given $M$ a finite von Neumann algebra with trace $\tau$, $T\in M$ invertible.

The Fuglede-Kadison determinant is defined as

$\Delta(T)=e^{\tau(log|T|)}$,

where $|T|=(T^*T)^{\frac{1}{2}}$, and $\tau(log|T|)=\int_{0}^{||T||}log(t)d\mu_{|T|}(t)$, and the probability measure $\mu_{|T|}$ is defined on spectrum(|T|) by requiring $\int_{spec(|T|)}fd\mu_{|T|}=\tau(f(|T|))$ for all $f\in C(spec(|T|))$.

See the reference Determinant theory in finite factors

Especially, let $M=L(\mathbb{F}_2)$ be the free group factor associated to the free group $\mathbb{F}_2$ on two generators $a, b$, and with the canonical trace $\tau$, my question is :

**1, Are there any references for the study of the determinant in the case $M=L(\mathbb{F}_2)$ ?**

Especially, I also want to know

**2, Are there any nontrivial computable examples in this case, i.e., what does $\Delta(T)$ looks like for $T\in \mathbb{C}\Gamma$, invertiable ?**

Note: This question is motivated by the paper Li, 2012.