A finite-entropy separation of microstates and nonmicrostates free entropy
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.
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On counterexamples to the unification of free entropyarXiv.orgDavid JekelOct 19, 2026in the index
It has been a long-standing open problem to determine when two versions of Voiculescu's free entropy, $χ$ and $χ^*$, agree. OpenAI recently generated the first example with $-\infty < χ< χ^*$. We give a detailed presentation of a similar example, and explicitly show the separation between free…