There is a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state for which both microstates and nonmicrostates free entropy are finite yet −∞<χ(X)≤χ*(X)−1/2<∞.
There is a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state for which both microstates and nonmicrostates free entropy are finite yet −∞<χ(X)≤χ*(X)−1/2<∞.
There is a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state for which both microstates and nonmicrostates free entropy are finite yet −∞<χ(X)≤χ*(X)−1/2<∞.
If correct, answers the finite-entropy equality question negatively with a quantitative gap.
Must finite microstates free entropy equal nonmicrostates free entropy?
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