Degree and arity of tracial separation for self-adjoint contractions

Abstract

Motivated by the recent explicit counterexample to Connes' embedding conjecture, we study the degree and number of variables needed to separate normalized matrix traces on self-adjoint contractions from traces on finite von Neumann algebras. We prove that the two trace infima coincide for every Hermitian polynomial of degree at most two in at most three variables, with a dimension-explicit approximation bound. We give three explicit separators: an integer-coefficient polynomial of degree six in ten variables with 392 nonzero words, a quadratic polynomial in 214 variables, and an integer-coefficient polynomial in two variables of ordinary and cyclic degree 24. Each has normalized trace at least 3/4 on every matrix contraction tuple and trace −1 at a finite tracial witness. We provide the source systems, transfer estimates, and exact coefficient recipes for all three constructions.

A degree-linear upper bound for the Pólya exponent of forms with fixed arity
Yulong Li, Colin Tan
arXiv10 Oct 2026PreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv
Weak tracial approximation and a weak Rokhlin property for C*-dynamical systems
George A. Elliott, Chun Guang Li, Zhuang Niu +1
arXiv27 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Relative entropy representations via tracial joint spectral measures
Lukas Schmitt
arXiv11 Sep 2026PreprintarXiv licenceFull textMathematical physics Quantum physics & quantum informationarXiv
Dimension-free Riesz transform estimates on biased hypercubes via non-tracial baby Fock models
Hun Hee Lee, Zhendong Xu, Sang-gyun Youn +1
arXiv21 Sep 2026PreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsarXiv
High-Arity Rademacher Complexity and Uniform Laws of Large Numbers
Aaron Anderson, Henry Towsner
arXiv13 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematics Logic & foundationsarXiv

Cite

BibTeX RIS

BibTeX

@misc{wang2026degree,
  title = {{Degree and arity of tracial separation for self-adjoint contractions}},
  author = {Jie Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02604},
  eprint = {2611.02604},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02604},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Degree and arity of tracial separation for self-adjoint contractions
AU  - Jie Wang
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02604
UR  - https://arxiv.org/abs/2611.02604
AB  - Motivated by the recent explicit counterexample to Connes' embedding conjecture, we study the degree and number of variables needed to separate normalized matrix traces on self-adjoint contractions from traces on finite von Neumann algebras. We prove that the two trace infima coincide for every Hermitian polynomial of degree at most two in at most three variables, with a dimension-explicit approximation bound. We give three explicit separators: an integer-coefficient polynomial of degree six in ten variables with 392 nonzero words, a quadratic polynomial in 214 variables, and an integer-coefficient polynomial in two variables of ordinary and cyclic degree 24. Each has normalized trace at least 3/4 on every matrix contraction tuple and trace −1 at a finite tracial witness. We provide the source systems, transfer estimates, and exact coefficient recipes for all three constructions.
ER  -

Abstract

Related

Cite