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On permanence of regularity properties II

Hyun Ho Lee

Abstract

In this article, we study the permanence of topological and algebraic dimension type properties of simple unital $C\sp*$-algebras. When a pair of unital $C\sp*$-algebras $(A, B)$ is associated by a $*$-homomorphism $φ: A\to B$ which is tracially sequentially-split by order zero, we show that under suitable assumptions on either $A$ or $B$, or both $A$ and $B$ \begin{itemize} \item tracial $m$-comparison passes from $B$ to $A$; \item tracial $m$-almost divisibility passes from $B$ to $A$; \item tracial nuclear dimension less than $m$ passes from $B$ to $A$. \end{itemize}

On permanence of regularity properties II

Abstract

In this article, we study the permanence of topological and algebraic dimension type properties of simple unital -algebras. When a pair of unital -algebras is associated by a -homomorphism which is tracially sequentially-split by order zero, we show that under suitable assumptions on either or , or both and \begin{itemize} \item tracial -comparison passes from to ; \item tracial -almost divisibility passes from to ; \item tracial nuclear dimension less than passes from to . \end{itemize}
Paper Structure (3 sections, 11 theorems, 76 equations)

This paper contains 3 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Proposition 2.3

For $a \in M_n(A)_+$ and $b \in M_m(A)_+$, the following are equivalent:

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2: Dimensional Independence in $M_{\infty}(A)$
  • Proposition 2.3: Rectangular Matrix Characterization
  • Definition 2.4: Cuntz Equivalence
  • Lemma 2.5
  • Definition 2.6: The Semigroup $W(A)$
  • Proposition 2.7: Well-definedness
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 18 more