Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The compact operators on $c_0$ as a Calkin algebra

Pavlos Motakis, Daniele Puglisi

Abstract

For a Banach space $X$, let $\mathcal{L}(X)$ denote the algebra of all bounded linear operators on $X$ and let $\mathcal{K}(X)$ denote the compact operator ideal in $\mathcal{L}(X)$. The quotient algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is called the Calkin algebra of $X$, and it is denoted $\mathcal{C}al(X)$. We prove that the unitization of $\mathcal{K}(c_0)$ is isomorphic as a Banach algebra to the Calkin algebra of some Banach space $\mathcal{Z}_{\mathcal{K}(c_0)}$. This Banach space is an Argyros-Haydon sum $(\oplus_{n=1}^\infty X_n)_\mathrm{AH}$ of a sequence of copies $X_n$ of a single Argyros-Haydon space $\mathfrak{X}_\mathrm{AH}$, and the external versus the internal Argyros-Haydon construction parameters are chosen from disjoint sets.

The compact operators on $c_0$ as a Calkin algebra

Abstract

For a Banach space , let denote the algebra of all bounded linear operators on and let denote the compact operator ideal in . The quotient algebra is called the Calkin algebra of , and it is denoted . We prove that the unitization of is isomorphic as a Banach algebra to the Calkin algebra of some Banach space . This Banach space is an Argyros-Haydon sum of a sequence of copies of a single Argyros-Haydon space , and the external versus the internal Argyros-Haydon construction parameters are chosen from disjoint sets.
Paper Structure (10 sections, 16 theorems, 61 equations)

This paper contains 10 sections, 16 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Horizontally approximable operators
  3. Bourgain-Delbaen-$\mathscr{L}_\infty$ sums of Banach spaces
  4. Argyros-Haydon sums of a sequence of Banach spaces
  5. Definition of the sets $(\Delta_n)_{n=1}^\infty$
  6. Definition of the space $\mathcal{Z} = \mathcal{Z}((m_j,n_j)_{j=1}^\infty,(X_n)_{n=1}^\infty)$
  7. Evaluation analysis of $e_\gamma^*$
  8. Operators on Argyros-Haydon sums of Banach spaces
  9. The space $\mathcal{Z}_{\mathcal{K}(c_0)}$
  10. Open problems and directions

Key Result

Theorem 1

There exists a Banach space $\mathcal{Z}_{\mathcal{K}(c_0)}$ with Calkin algebra isomorphic to the unitization of $\mathcal{K}(c_0)$ as a Banach algebra.

Theorems & Definitions (38)

  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Definition 2.8
  • Proposition 2.9
  • proof
  • ...and 28 more