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Approximate identities for Ideals in $L(L^p))$

William B. Johnson, Gideon Schechtman

Abstract

The main result is that the only non trivial closed ideal in the Banach algebra $L(L^p)$ of bounded linear operators on $L^p(0,1)$, $1\le p < \infty$, that has a left approximate identity is the ideal of compact operators. The algebra $L(L^1)$ has at least one non trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.

Approximate identities for Ideals in $L(L^p))$

Abstract

The main result is that the only non trivial closed ideal in the Banach algebra of bounded linear operators on , , that has a left approximate identity is the ideal of compact operators. The algebra has at least one non trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.
Paper Structure (3 sections, 7 theorems, 24 equations)

This paper contains 3 sections, 7 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Left approximate identities for ideals in $L(L^p)$
  3. Right approximate identities for ideals in $L(L^1)$

Key Result

Theorem 2.1

Let $\mathcal{I}$ be a non trivial closed ideal in $L(L^p)$, $1\le p < \infty$, and assume that $\mathcal{I}$ has a left approximate identity. Then $\mathcal{I}$ is ${\mathcal{K}}(L^p)$, the ideal of compact operators.

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Theorem : 2.3.4 in mu
  • Theorem 3.1
  • Claim 3.2
  • Lemma 3.3
  • Theorem 3.4