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](http://arxiv.org/abs/1001.0419)


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](http://arxiv.org/abs/1001.0419). 



Fuglede-Kadison determinants in $L(\mathbb{F}_2)$
The spectral measures for self-adjoint elements in $\mathbb C F\_2$ are very special. In particular, it is known that non of the elements in $\mathbb C F\_2$ has a kernel when acting via the left-regular representation on $\ell^2 F\_2$. This was shown by Peter Linnell using index-theoretic methods in


Linnell, Peter,
*Division rings and group von Neumann algebras.*
Forum Math. 5 (1993), no. 6, 561–576. 


(It also follows from another of Linnell's papers and the fact that $F\_2$ is left-orderable.)


It is also known that Novikov-Shubin invariants are always positive. This together implies that $\Delta(T) \neq 0$ if $T\neq 0$. 


The main advantage is that $\sum\_{n} \tau(T^n)z^n$ is an algebraic power series for any $T \in \mathbb C F\_2$. This implies the result about Novikov-Shubin invariants (and much more). It was proved in


Sauer, Roman, 
*Power series over the group ring of a free group and applications to Novikov-Shubin invariants.* High-dimensional manifold topology, 449–468, World Sci. Publ., River Edge, NJ, 2003


Starting with the computations in this paper (and using the Stieltjes-Inversion formula), you can also make concrete computations for the determinant of specific elements.


