Algebra · Oct 6, 2026

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.

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