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 II₁ 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 ℂⁿ. 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 ℂⁿ. 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 II₁ factor. In one variable, the representing measure is simply the Riesz measure of one transformed function. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.