Group-Ring Determinant Conjecture

The group-ring determinant conjecture asserts a nonnegative logarithmic determinant for every matrix over the integral group ring of every discrete group. For a square matrix , let act by left convolution on . The spectral theorem gives a probability measure on characterized by

(1)

for continuous functions on the operator spectrum, where is the unit vector supported at the identity element in coordinate . The conjectured inequality for these square matrices is

(2)

The mass at 0 is omitted.

For invertible , the Fuglede-Kadison determinant with the unnormalized matrix extension of the canonical group von Neumann algebra trace satisfies

(3)