Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Diagonal comparison of ample C*-diagonals

Grigoris Kopsacheilis, Wilhelm Winter

TL;DR

The paper introduces diagonal comparison for ample C*-diagonal pairs (D⊆A), defining a diagonally implemented subequivalence that strengthens ordinary Cuntz subequivalence via a normaliser-driven factorisation. It proves that diagonal comparison is equivalent to the conjunction of dynamical comparison and strict comparison under hereditary assumptions, linking topological dynamics to C*-algebra regularity. A key intermediate property, tracial almost divisibility, enables trace-wise interpolation essential to the main argument and yields corollaries for transformation groupoids, including finite diagonal dimension implying tracial Z-stability. The work also studies when the conditional expectation associated to such pairs is hereditary, showing finite diagonal dimension guarantees heredity, while identifying natural counterexamples where heredity fails, and discussing open questions about automatic heredity in broader settings.

Abstract

We introduce diagonal comparison, a regularity property of diagonal pairs where the sub-C*-algebra has totally disconnected spectrum, and establish its equivalence with the concurrence of strict comparison of the ambient C*-algebra and dynamical comparison of the underlying dynamics induced by the partial action of the normalisers. As an application, we show that for diagonal pairs arising from principal minimal transformation groupoids with totally disconnected unit space, diagonal comparison is equivalent to tracial Z-stability of the pair and that it is implied by finite diagonal dimension. In-between, we show that any projection of the diagonal sub-C*-algebra can be uniformly tracially divided, and explore a property of conditional expectations onto abelian sub-C*-algebras, namely containment of every positive element in the hereditary subalgebra generated by its conditional expectation. We show that the expectation associated to a C*-pair with finite diagonal dimension is always hereditary in that sense, and we give an example where this property does not occur.

Diagonal comparison of ample C*-diagonals

TL;DR

The paper introduces diagonal comparison for ample C*-diagonal pairs (D⊆A), defining a diagonally implemented subequivalence that strengthens ordinary Cuntz subequivalence via a normaliser-driven factorisation. It proves that diagonal comparison is equivalent to the conjunction of dynamical comparison and strict comparison under hereditary assumptions, linking topological dynamics to C*-algebra regularity. A key intermediate property, tracial almost divisibility, enables trace-wise interpolation essential to the main argument and yields corollaries for transformation groupoids, including finite diagonal dimension implying tracial Z-stability. The work also studies when the conditional expectation associated to such pairs is hereditary, showing finite diagonal dimension guarantees heredity, while identifying natural counterexamples where heredity fails, and discussing open questions about automatic heredity in broader settings.

Abstract

We introduce diagonal comparison, a regularity property of diagonal pairs where the sub-C*-algebra has totally disconnected spectrum, and establish its equivalence with the concurrence of strict comparison of the ambient C*-algebra and dynamical comparison of the underlying dynamics induced by the partial action of the normalisers. As an application, we show that for diagonal pairs arising from principal minimal transformation groupoids with totally disconnected unit space, diagonal comparison is equivalent to tracial Z-stability of the pair and that it is implied by finite diagonal dimension. In-between, we show that any projection of the diagonal sub-C*-algebra can be uniformly tracially divided, and explore a property of conditional expectations onto abelian sub-C*-algebras, namely containment of every positive element in the hereditary subalgebra generated by its conditional expectation. We show that the expectation associated to a C*-pair with finite diagonal dimension is always hereditary in that sense, and we give an example where this property does not occur.
Paper Structure (4 sections, 20 theorems, 41 equations)

This paper contains 4 sections, 20 theorems, 41 equations.

Table of Contents

  1. Diagonal comparison
  2. Tracial almost divisibility
  3. Main result and consequences
  4. Hereditary conditional expectations

Key Result

Theorem B

Let $(D \subset A)$ be a unital, ample $\mathrm{C}^\ast$-diagonal pair with hereditary associated conditional expectation,A conditional expectation $A\to D$ is called hereditary when each positive element of $A$ lies in the hereditary sub-$\mathrm{C}^\ast$-algebra generated by its image through the

Theorems & Definitions (48)

  • Definition A
  • Theorem B
  • Corollary C
  • Theorem D
  • Proposition E
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • ...and 38 more