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Math-2378 · mathematics · Algebra

$\mathcal Z$-Stability and Regularity of Nuclear $C^*$-Algebras

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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.
Does $\mathcal Z$-stability imply finite nuclear dimension for all unital simple separable nuclear $C^*$-algebras? More broadly, does the equivalence between $\mathcal Z$-stability and strict comparison hold without the additional hypotheses currently known?

Formulation references: [S1], [S2].

Short English statement

Does $\mathcal Z$-stability imply finite nuclear dimension for all unital simple separable nuclear $C^*$-algebras? More broadly, does the equivalence between $\mathcal Z$-stability and strict comparison hold without the additional hypotheses currently known?

Sources

Catalogue entry

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

Contributions 1

Math-2378-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2378

Manuscript 1 (family 291, 2026-09-23): The paper resolves regularity for the specified separable nonelementary or infinite-dimensional setting, but the broader strict-comparison question in this bundle supplies no such complete hypothesis restriction.

Manuscript 2 (family 291, 2026-09-23): The paper resolves regularity for the specified separable nonelementary or infinite-dimensional setting, but the broader strict-comparison question in this bundle supplies no such complete hypothesis restriction.

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mathcal Z\$-Stability and Regularity of Nuclear \$C^*\$-Algebras

OpenAI's claim

Manuscript 1 (family 291, 2026-09-23): Finite nuclear dimension is claimed to characterize Jiang--Su stability for separable nuclear algebras lacking elementary ideal subquotients; moreover, Jiang--Su stabilization always lowers nuclear dimension to at most one. Manuscript 2 (family 291, 2026-09-23): The paper claims uniform Gamma from real rank zero in the tracial-completion ultrapower for simple unital nuclear stably finite algebras. Comparison gives absorption, also under specified bounded-trace hypotheses in the stably projectionless setting.

Mathematical statement of the claimed result

Mathematical claim 1.
For a separable nuclear complex $C^*$-algebra $A$ with no ideal subquotient that is nonzero and elementary $I/J\simeq\mathcal K(H)$, OpenAI claims \[ \dim_{\mathrm{nuc}}A<\infty\quad\Longleftrightarrow\quad A\simeq A\otimes_{\min}\mathcal Z\quad\Longleftrightarrow\quad\dim_{\mathrm{nuc}}A\le1. \] Nuclear dimension is defined by finite-dimensional completely positive approximations with order-zero color decomposition; $\mathcal Z$ is the Jiang--Su algebra. The conditions are also claimed equivalent to either a canonical or an abstract isomorphism of $\operatorname{Cu}(A)$ with $\operatorname{Cu}(A\otimes_{\min}\mathcal Z)$. Without the subquotient restriction it asserts $\dim_{\mathrm{nuc}}(A_0\otimes_{\min}\mathcal Z)\le1$ for every separable nuclear $A_0$. Separately, for every nonzero unital characterless $D$ it claims a finite maximal tensor power containing two full orthogonal positive elements; full means generating the whole algebra as a closed ideal.
Mathematical claim 2.
Let $A$ be unital, separable, simple, infinite-dimensional, nuclear and stably finite, with nonempty tracial state space $T(A)$. Let $B=\overline A^{T(A)}$ be its completion on the norm-bounded unit ball for $\|a\|_{2,T(A)}=\sup_\tau\tau(a^*a)^{1/2}$, and $B^\omega$ its uniform-tracial ultrapower for a free ultrafilter. If $B^\omega$ has real rank equal to zero when given the quotient $C^*$-norm, OpenAI claims a central projection $p$ satisfying \[ \lambda(px)=\tfrac12\lambda(x)\qquad(x\in B,\ \lambda\in\Lambda_\omega), \] where $\Lambda_\omega$ consists of ultralimits of sequences of designated trace extensions. It also claims every tracial state on $B$ is the unique extension of a trace of $A$, with $|\sigma(b)|\le\|b\|_{2,T(A)}$. For a unital nuclear inclusion $A\subset D$ with $D$ real rank zero and nonempty traces, assume arbitrarily small uniformly tracial full projections in $D$. It claims finite-set approximate central halving projections: given finite $K\subset\mathcal U(A)$ and $\varepsilon>0$, some projection $P\in D$ satisfies $\sup_\tau\|[P,x]\|_{2,\tau}<\varepsilon$ and $\sup_\tau|\tau(Px)-\tau(x)/2|<\varepsilon$ for $x\in K$. The independent unital result takes an infinite-dimensional nuclear complex $C^*$-algebra $A$, assumed simple, separable and unital. For positive $a,b\in A\otimes\mathcal K$, define $d_\tau(a)=\lim_{n\to\infty}(\tau\otimes\operatorname{Tr})(a^{1/n})$, using the unnormalized matrix trace. Assume \[ b\ne0,\quad d_\tau(a)<d_\tau(b)\quad(\tau\in T(A))\quad\Longrightarrow\quad a\precsim b. \] When $T(A)$ is empty, this premise still requires $a\precsim b$ for every positive $a$ and every nonzero positive $b$. OpenAI claims $A\simeq A\otimes\mathcal Z$; when traces exist it also constructs uniform property $\Gamma$. The separate nonunital theorem starts with a nonzero nuclear complex $C^*$-algebra $A$ that is simple and separable, with no nonzero projection in $A\otimes\mathcal K$. Write $T_{\mathrm{lsc}}(A)$ for densely finite lower-semicontinuous traces, and $T_1(A)$ for those bounded traces having norm one. Require every trace in $T_{\mathrm{lsc}}(A)$ to be bounded on the positive unit ball, and require $T_1(A)$ to be nonempty and weak-$*$ compact. The precise comparison premise is \[ \forall a,b\in(A\otimes\mathcal K)_+,\quad b\ne0,\quad \left[d_\tau(a)<1\text{ for every }\tau\in T_{\mathrm{lsc}}(A)\text{ with }d_\tau(b)=1\right] \ \Longrightarrow\ a\precsim b. \] Under these hypotheses it claims uniform property $\Gamma$ and $A\simeq A\otimes\mathcal Z$. Here $a\precsim b$ means $x_n^*bx_n\to a$ in norm for stabilized elements $x_n$, and $\mathcal Z$ is the Jiang--Su algebra.

Outline of the argument

Manuscript 1 (family 291, 2026-09-23): Localized order-zero approximations control ranks even when traces are infinite on some ideals. A supporting map and comparison yield two-color approximations that survive stabilization and hereditary passage. Conversely, Gaussian sign tensors and pure-state excision construct two full orthogonal central elements from the absence of characters, supplying the absorption criterion.

Manuscript 2 (family 291, 2026-09-23): Nuclear averaging and scalar block compressions preserve coefficient traces while packing small projections with controlled commutators. A separate comparison construction rounds ranks, forms clipped sums of centered symmetries, and extracts Gaussian trace partitions. In the projectionless case, a combined overlap-and-model cost controls small splittings before reindexing amplifies them to exact central partitions.

Formal verification

For family 291, the scope page documents the following selections. The selection gives uniform Gamma from real rank zero of the designated tracial ultrapower. The documented comparison scope also treats Jiang--Su absorption and Gamma under strict comparison, with additional trace and target-rank conditions in the projectionless case. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. The documented statement covers the main claim of manuscript 2. Comparator files: UniformGamma.lean. Declarations: OAI.ComparatorModel.CurrentMain.main.

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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