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Math-2437 · mathematics · Algebra

Nuclear Dimension of Transformation-Group Crossed Products

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a continuous action of a locally compact group $G$ on a locally compact Hausdorff space $X$, when does the crossed product $C_0(X)\rtimes G$ have finite nuclear dimension?

Formulation references: [S1], [S2].

Short English statement

For a continuous action of a locally compact group $G$ on a locally compact Hausdorff space $X$, when does the crossed product $C_0(X)\rtimes G$ have finite nuclear dimension?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2437-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2437

Manuscript 1 (family 302, 2026-09-25): For minimal Z actions on infinite compact metrizable spaces, rc=mdim/2 and the zero-dimension corollary determine finite nuclear dimension; this problem asks a general locally compact-group transformation-group classification, including nonminimal and nonmetrizable cases.

Nuclear Dimension of Transformation-Group Crossed Products

OpenAI's claim

Whenever an infinite compact metrizable system is minimal under a homeomorphism, its crossed-product comparison radius is claimed to be half its mean dimension, including infinite values; zero forces nuclear dimension at most one. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $h$ be a homeomorphism of $X$ whose orbits are all dense, where $X$ is compact, infinite and metrizable. OpenAI claims \[ \operatorname{rc}(C(X)\rtimes_h\mathbb Z)=\tfrac12\operatorname{mdim}(X,h)\quad\text{in }[0,\infty]. \] For positive stabilized elements define $d_\tau(a)=\lim_m\tau(a^{1/m})$. The radius of comparison is the infimum of $r\ge0$ such that $d_\tau(a)+r<d_\tau(b)$ for every normalized trace implies Cuntz subequivalence $a\precsim b$. Mean dimension is the asymptotic covering dimension per iterate, defined from joins of finite open covers. Both finite and infinite mean dimensions are included.

Outline of the argument

Manuscript 1 (family 302, 2026-09-25): Finite-support comparison witnesses are converted to bundle embeddings, which feed the companion's boundary-preserving compression theorem. Packet reconstruction translates that compression into a smaller cover without introducing absent colors. Applying the construction on aligned orbit blocks contradicts any larger cover-growth rate, providing the missing lower bound; established upper bounds and regularity results finish the equality and zero-dimensional consequences.

Formal verification

Family 302 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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