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Math-2831 · mathematics · Analysis

Uniform Gamma from tracial-ultrapower real rank zero

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let A be simple separable unital infinite-dimensional nuclear stably finite with nonempty tracial-state space, and let B be its uniform tracial completion. For any free ultrafilter $\omega $, does real rank zero of the designated uniform tracial ultrapower $B^\omega $ in its quotient C*-norm imply uniform property Gamma for A?

Short English statement

Let A be simple separable unital infinite-dimensional nuclear stably finite with nonempty tracial-state space, and let B be its uniform tracial completion. For any free ultrafilter $\omega $, does real rank zero of the designated uniform tracial ultrapower $B^\omega $ in its quotient C*-norm imply uniform property Gamma for A?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2831-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2831

Manuscript 1 (family 291, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Uniform Gamma from tracial-ultrapower real rank zero

OpenAI's claim

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. 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 $A$ 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): 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. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. 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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