A Slice–Riesz–Bochner Characterization of Joint Brown Determinant Functions

We characterize the functions arising as Fuglede–Kadison determinant transforms of bounded commuting tuples in a fixed \(\mathrm{II}_1\) factor. After recovering the log-modulus formula intended by an AIM range question, we show that such functions are exactly the logarithmic potentials of compact probability measures on \(\mathbb C^n\) . Intrinsically, each complex line must give a normalized subharmonic logarithmic potential with uniformly bounded Riesz support, and the unit-frequency transforms of the slice measures must form a continuous positive-definite function on \(\mathbb C^n\) . Complex scaling of all slice measures is forced automatically by the single global function; Bochner's theorem then reconstructs the unique joint measure. Every compact probability measure is realized by a commuting normal tuple in any prescribed \(\mathrm{II}_1\) factor. In one variable, the representing measure is simply the Riesz measure of one transformed function.