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On $\ell$-open $C^*$ algebras and $\ell$-closed $C^*$-algebras

Dolapo Oyetunbi, Aaron Tikuisis

Abstract

In this paper, we characterize $\ell$-open and $\ell$-closed $C^*$-algebras and deduce that $\ell$-open $C^*$-algebras are $\ell$-closed, as conjectured by Blackadar. Moreover, we show that a commutative unital $C^*$-algebra is $\ell$-open if and only if it is semiprojective.

On $\ell$-open $C^*$ algebras and $\ell$-closed $C^*$-algebras

Abstract

In this paper, we characterize -open and -closed -algebras and deduce that -open -algebras are -closed, as conjectured by Blackadar. Moreover, we show that a commutative unital -algebra is -open if and only if it is semiprojective.
Paper Structure (5 sections, 15 theorems, 25 equations)

This paper contains 5 sections, 15 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties and characterization of $\ell$-open $C^*$-algebras
  4. Characterization of $\ell$-closed $C^*$-algebras
  5. Commutative unital $\ell$-open $C^*$-algebras

Key Result

Theorem 1.1

Let $A$ be a $C^*$-algebra. The following are equivalent:

Theorems & Definitions (34)

  • Theorem 1.1: see Theorem \ref{['thm:Main']}
  • Theorem 1.2: see Theorem \ref{['thm:MainClosed']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4: Bla16, Proposition 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2: cf. Bla16
  • proof
  • ...and 24 more