Two notions of free entropy differ even when both are finite. Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $-\infty\lt \chi\lt \chi^*\lt \infty$. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.
released 2026-09-25 | 1 theorem · 7 lemmas · 9 proofs · 6,133 words |
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We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that $-\infty\lt \chi(X)\leq\chi^*(X)-\tfrac12\lt \infty$. Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.