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Tracial approximation and ${\cal Z}$-stability

Huaxin Lin

TL;DR

This work advances the classification program for separable amenable simple C*-algebras by proving that, for non-elementary algebras with a tracial cone $ ilde T(A)$ whose extremal boundary is $\sigma$-compact and countable-dimensional, ${\cal Z}$-stability is equivalent to strict comparison together with stable rank one (or, equivalently, with strict comparison plus the $T$-tracial approximate oscillation zero condition). The authors leverage a trace-norm/ultrapower framework to perform tracial approximations directly in $l^{\infty}(A)/I_{\varpi}$, avoiding the central sequence algebra, and develop a suite of tools—semi-projectivity in 2-norm, finite-dimensional 2-norm approximations, and tracial commutativity results—to build ${\cal Z}$-stability from (and only from) these trace-analytic hypotheses. The core contribution is a unifying theorem that extends prior results to non-Bauer trace spaces with infinite-dimensional, countable-dimensional extremal boundaries and to non-unital (including projectionless) algebras, thereby broadening the class of algebras for which the Toms–Winter equivalences hold. This broadens the scope of Elliott-type classification, showing that stable rank and trace-geometry conditions suffice to guarantee ${\cal Z}$-stability in a wide array of settings, with potential implications for regularity, McDuff-type properties, and future WTAC-driven classification programs.

Abstract

Let $A$ be a unital separable non-elementary amenable simple stably finite C*-algebra such that its tracial state space has a $σ$-compact countable-dimensional extremal boundary. We show that $A$ is ${\cal Z}$-stable if and only if it has strict comparison and stable rank one. We show that this result also holds for non-unital cases (which may not be Morita equivalent to unital ones).

Tracial approximation and ${\cal Z}$-stability

TL;DR

This work advances the classification program for separable amenable simple C*-algebras by proving that, for non-elementary algebras with a tracial cone whose extremal boundary is -compact and countable-dimensional, -stability is equivalent to strict comparison together with stable rank one (or, equivalently, with strict comparison plus the -tracial approximate oscillation zero condition). The authors leverage a trace-norm/ultrapower framework to perform tracial approximations directly in , avoiding the central sequence algebra, and develop a suite of tools—semi-projectivity in 2-norm, finite-dimensional 2-norm approximations, and tracial commutativity results—to build -stability from (and only from) these trace-analytic hypotheses. The core contribution is a unifying theorem that extends prior results to non-Bauer trace spaces with infinite-dimensional, countable-dimensional extremal boundaries and to non-unital (including projectionless) algebras, thereby broadening the class of algebras for which the Toms–Winter equivalences hold. This broadens the scope of Elliott-type classification, showing that stable rank and trace-geometry conditions suffice to guarantee -stability in a wide array of settings, with potential implications for regularity, McDuff-type properties, and future WTAC-driven classification programs.

Abstract

Let be a unital separable non-elementary amenable simple stably finite C*-algebra such that its tracial state space has a -compact countable-dimensional extremal boundary. We show that is -stable if and only if it has strict comparison and stable rank one. We show that this result also holds for non-unital cases (which may not be Morita equivalent to unital ones).
Paper Structure (9 sections, 48 theorems, 260 equations)

This paper contains 9 sections, 48 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Some basics and quotients
  4. Examples of Choquet simplices and separation
  5. Tracial approximate commutativity
  6. Semi-projectivity in 2-norm
  7. Finite dimensional approximation in 2-norm
  8. Countable dimensional extremal boundaries
  9. Epilogue

Key Result

Theorem 1.1

Let $A$ be a non-elementary separable amenable simple $C^*$-algebra with $\widetilde{T}(A)\setminus \{0\}\not=\emptyset$ such that $\widetilde{T}(A)$ has a $\sigma$-compact countable-dimensional extremal boundary (see Definition Dscdim). Then the following are equivalent. (1) $A$ has strict compari

Theorems & Definitions (114)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 104 more