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Tracial oscillation zero and Z-stability

Huaxin Lin

Abstract

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal Z}$-stable if and only if $A$ has strict comparison for positive elements. Extremal boundaries of simplexes which satisfy condition (C) may contain countable disjoint unions of $n$-dimensional cubes ($n\in \N$) as a subset.

Tracial oscillation zero and Z-stability

Abstract

Let be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that is -stable if and only if has strict comparison for positive elements. Extremal boundaries of simplexes which satisfy condition (C) may contain countable disjoint unions of -dimensional cubes () as a subset.
Paper Structure (7 sections, 30 theorems, 215 equations)

This paper contains 7 sections, 30 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Extremal Boundaries
  4. Oscillation and orthogonal complements
  5. Lifting
  6. Sums of order zero maps
  7. The main results

Key Result

Proposition 2.14

If $S$ is a compact subset of $\partial_e(T),$ then $\overline{{\rm conv}(S)}=M_S.$

Theorems & Definitions (80)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • ...and 70 more