  Motivated by our results in "Polish Models and Sofic Entropy," we define
modified version of the independence tuples for sofic entropy developed by Kerr
and Li. These modified version essentially require that the independence
sequences give rise to representations weakly contained in the left regular
when projected onto the Koopman representation. Using this, we can generalize
our previous results for Deninger's Problem. Namely, we can show that if G is a
sofic group, and if f is in M_{n}(Z(G)) and is invertible as an operator on
l^{2}(G)^{n}, then the Fuglede-Kadison determinant of f is 1 if and only if f
is invertible in M_{n}(Z(G)).
