Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$C(K)$-spaces with few operators relative to posets

Antonio Acuaviva

TL;DR

The paper develops a framework to build nonseparable C(K)-spaces with few operators organized by a poset P, using almost disjoint families to create C0(K_A) blocks that embed into larger spaces. It proves that operators between blocks decompose into a scalar part plus a separable-range part when the index respects the poset, while noncomparable indices force separable-range behavior, enabling a complete classification of closed operator ideals in certain quotient algebras. By introducing incidence algebras and c0-incidence algebras, the authors realize a wide class of unital Banach algebras as quotients of operator algebras, and demonstrate strongly split exact sequences connecting these quotients to block-structured operator algebras. This yields both a broad mechanism to engineer operator-ideal lattices (including non-linearly ordered ones) and new results on automatic continuity in the CH setting. The constructions culminate in a general quotient-algebras theorem, showing that any sequence of algebras can be realized as quotients of a single B(Y)/X(Y) for a suitably chosen C0(K_A) space.

Abstract

Extending a method developed by Koszmider and Laustsen for constructing $C(K)$-spaces we produce families of $C(K)$-spaces with few operators relative to a partially ordered set $\mathcal{P}$. Using these spaces, we construct new $C(K)$-spaces whose closed operator ideals can be completely classified. Additionally, we use these spaces to resolve some questions regarding automatic continuity.

$C(K)$-spaces with few operators relative to posets

TL;DR

The paper develops a framework to build nonseparable C(K)-spaces with few operators organized by a poset P, using almost disjoint families to create C0(K_A) blocks that embed into larger spaces. It proves that operators between blocks decompose into a scalar part plus a separable-range part when the index respects the poset, while noncomparable indices force separable-range behavior, enabling a complete classification of closed operator ideals in certain quotient algebras. By introducing incidence algebras and c0-incidence algebras, the authors realize a wide class of unital Banach algebras as quotients of operator algebras, and demonstrate strongly split exact sequences connecting these quotients to block-structured operator algebras. This yields both a broad mechanism to engineer operator-ideal lattices (including non-linearly ordered ones) and new results on automatic continuity in the CH setting. The constructions culminate in a general quotient-algebras theorem, showing that any sequence of algebras can be realized as quotients of a single B(Y)/X(Y) for a suitably chosen C0(K_A) space.

Abstract

Extending a method developed by Koszmider and Laustsen for constructing -spaces we produce families of -spaces with few operators relative to a partially ordered set . Using these spaces, we construct new -spaces whose closed operator ideals can be completely classified. Additionally, we use these spaces to resolve some questions regarding automatic continuity.

Paper Structure

This paper contains 11 sections, 29 theorems, 56 equations.

Table of Contents

  1. Introduction and main results
  2. Organisation and notation
  3. Preliminary results
  4. Structure and combination of .
  5. An overview of matrices acting on almost disjoint families.
  6. Operators and their matrix representation.
  7. Proof of Theorem \ref{['th: almost-disjoint-families-few-operators']}
  8. Quotient algebras and operator ideals
  9. Finite-dimensional incidence algebras
  10. $c_0$-Incidence algebras.
  11. Proof of Theorem \ref{['th: quotient-algebras']}.

Key Result

Theorem 1.1

Suppose $\mathcal{P}$ is a partially ordered set of cardinality at most continuum. Then there exist a collection $(\mathcal{A}_t)_{t \in \mathcal{P}}$ of uncountable, almost disjoint families $\mathcal{A}_t \subseteq \left[{\mathbb{N}}\right]^{\omega}$ and a collection of isometric algebra homomorph

Theorems & Definitions (74)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • proof
  • Remark 1.7
  • Remark 1.8
  • Proposition 1.9
  • ...and 64 more