Independence Tuples and Deninger's Problem

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)).