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Continuous functions over a pure C*-algebra

Apurva Seth, Eduard Vilalta

Abstract

Let $X$ be a compact metric space, and let $A$ be a pure $\mathrm{C}^*$-algebra. We show that $C(X,A)$ is pure whenever $A$ is simple; or every quotient of $A$ is stably finite (e.g., $A$ has stable rank one). Using permanence properties of pureness, we prove that the tensor product of any such $A$ with any ASH-algebra is pure.

Continuous functions over a pure C*-algebra

Abstract

Let be a compact metric space, and let be a pure -algebra. We show that is pure whenever is simple; or every quotient of is stably finite (e.g., has stable rank one). Using permanence properties of pureness, we prove that the tensor product of any such with any ASH-algebra is pure.
Paper Structure (6 sections, 20 theorems, 92 equations)

This paper contains 6 sections, 20 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Comparison in Cu(C(X,A))
  4. Divisibility in Cu(C(X,A))
  5. Dimension reduction
  6. Pureness of Recursive Subhomogeneous Algebras

Key Result

Theorem A

Let $X$ be a compact metric space, and let $A$ be a pure $\mathrm{C}^*$-algebra. Assume additionally that Then, $C(X,A)$ is pure.

Theorems & Definitions (44)

  • Theorem A: \ref{['Purehomogenous']}, \ref{['prp:MainNonSimp']}
  • Theorem B: \ref{['prp:ASHtenPure']}
  • Definition 2.1: Coward
  • Lemma 2.3
  • proof
  • Definition 3.2: Win12NuclDimZstable
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 34 more