The Fuglede-Kadison determinant of an invertible operator 
in a von Neumann algebra with a faithful normal
finite trace 
is
 |
The natural logarithm of the positive operator 
is defined using the spectral
theorem (de la Harpe 2012). The value depends on the trace normalization. For
an invertible complex matrix 
and the ordinary matrix trace,
it is 
.
For the normalized matrix trace 
, it is 
. For example, 
gives 6 with the ordinary matrix
trace and 
with the normalized matrix
trace.
The group-ring determinant conjecture uses the canonical trace on group von Neumann algebra matrices, extended by summing the traces of the diagonal entries. For operators with a nonzero linear transformation kernel, its formulation discards the zero spectral mass. This requires a separate convention from the invertible formula above.
