Lower Semicontinuity of Free Entropy Dimension

Definitions and mathematical statement

Let $X^{(j)}=(X_1^{(j)},\ldots,X_n^{(j)})$ be tuples converging to $X=(X_1,\ldots,X_n)$ in the source's intended topology, and let $\delta$ be its intended free entropy dimension. The exact proposed lower-semicontinuity statement is \[ \delta(X_1,\ldots,X_n)\leq \liminf_{j\to\infty}\delta(X_1^{(j)},\ldots,X_n^{(j)}). \] It is to hold for all admissible convergent sequences and all allowed tuple sizes, after their common domain and convergence notion have been fixed. The inequality direction is part of the question; it is not upper semicontinuity.

Free entropy dimensions are ordinarily attached to tuples in a tracial noncommutative probability space, commonly self-adjoint elements of a finite von Neumann algebra $(\mathcal M,\tau)$. Several distinct invariants occur in this setting, including the original microstates free entropy dimension, modified microstates dimension $\delta_0$, and nonmicrostates variants. The source writes only $\delta$ and reproduces no definition, so none of these alternatives is silently substituted or identified with the others.

The mode of convergence is also absent. Strong-operator convergence of uniformly norm-bounded tuples in one algebra and convergence in $*$-distribution, meaning convergence of all tracial noncommutative polynomial moments, are different hypotheses. Allowing tuples in different algebras, imposing uniform operator-norm bounds, and requiring microstates or embeddability can affect the precise statement and its domain. The draft does not choose the weakest or strongest of these possibilities. A self-contained definition of the invariant, its allowed values, the tracial setting, and the intended convergence are required before the displayed inequality is a fully specified conjecture.

The supplied literature assessment checked 2026-08-21 explicitly identifies insufficient context for a claimed resolution and reports that no general proof was located. Its open label is a dated curation assessment of this extracted question, not a verified assertion for every possible variant and topology.

Short English statement

Does free entropy dimension at the limit stay below the lower limit of the dimensions of approximating tuples? The source omits both the version of free entropy dimension and the convergence topology, which are essential to the question.

Sources

• Local UnsolvedMath registry, permanent ID 30000316, catalogue entry, Lower Semicontinuity of Free Entropy Dimension.
• C*-Algebren (2006), Oberwolfach Reports, pp. 2305--2374, DOI https://doi.org/10.4171/owr/2005/41.
• Supplied literature triage checked 2026-08-21 cites only the original report; no separate definition, theorem, or convergence hypothesis is supplied.