Question

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.

Answer 1

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.