Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Way-below Relation and Tensor Products

Cristian Ivanescu, Hunter Labrecque

TL;DR

This work studies how the way-below relation, a refined form of containment in the sense of compact containment, behaves under tensor products in the Cuntz semigroup. It develops a noncommutative framework via countably generated Hilbert C*-modules and Kasparov stabilization, and proves that compact containment is preserved under tensor products under standard hypotheses (stable rank one, separable nuclear algebras, and sigma-unital hereditary subalgebras). The authors provide two proofs of the main result: an epsilon-cutdown approach and an argument based on Gardella–Perera's commutative characterization, plus a commutative reduction using open supports on product spaces. Appendices connect these noncommutative concepts to the classical theory of ideals in C(X) and their relation to open subsets, highlighting the duality between ideals and topological data in the commutative case.

Abstract

We establish the preservation of the way-below relation with respect to the tensor product.

Way-below Relation and Tensor Products

TL;DR

This work studies how the way-below relation, a refined form of containment in the sense of compact containment, behaves under tensor products in the Cuntz semigroup. It develops a noncommutative framework via countably generated Hilbert C*-modules and Kasparov stabilization, and proves that compact containment is preserved under tensor products under standard hypotheses (stable rank one, separable nuclear algebras, and sigma-unital hereditary subalgebras). The authors provide two proofs of the main result: an epsilon-cutdown approach and an argument based on Gardella–Perera's commutative characterization, plus a commutative reduction using open supports on product spaces. Appendices connect these noncommutative concepts to the classical theory of ideals in C(X) and their relation to open subsets, highlighting the duality between ideals and topological data in the commutative case.

Abstract

We establish the preservation of the way-below relation with respect to the tensor product.

Paper Structure

This paper contains 7 sections, 6 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Non-commutative definition of compact containment
  3. Way-below relation and tensor products: the commutative case
  4. Main Result
  5. Compact containment in the Cuntz semigroup
  6. Appendix A
  7. Appendix B

Key Result

Proposition 3.1

Let $I_1$ and $I_2$ be closed ideals in $C(K)$, compactly contained in $J_1$ and $J_2$, respectively. Then $I_1 \otimes I_2$ is compactly contained in $J_1 \otimes J_2$.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Definition 2.4
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • ...and 8 more