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$L^p$-modules and $L^p$-correspondences

Alonso Delfín

Abstract

We introduce an $L^p$-operator algebraic analogue of Hilbert C*- modules. We present the theory of concrete $L^p$-modules, their morphisms, and basic constructions including countable direct sums and tensor products. We then define $L^p$-correspondences and the interior tensor product of these.

$L^p$-modules and $L^p$-correspondences

Abstract

We introduce an -operator algebraic analogue of Hilbert C*- modules. We present the theory of concrete -modules, their morphisms, and basic constructions including countable direct sums and tensor products. We then define -correspondences and the interior tensor product of these.

Paper Structure

This paper contains 13 sections, 8 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^p$-operator algebras.
  4. Spatial Tensor Product
  5. $L^p$-modules over $L^p$-operator algebras
  6. $L^p$-modules and C*-like $L^p$-modules
  7. Finite Direct Sum of $L^p$-modules
  8. Countable Direct Sums of $L^p$-modules
  9. External Tensor Product of $L^p$-modules
  10. Morphisms of $L^p$-modules
  11. $L^p$-correspondences
  12. $L^p$-correspondences
  13. Tensor Product of $L^p$-correspondences

Key Result

Theorem 3.13

Let $(\mathsf{Y}, \mathsf{X}) = \bigoplus_{j = 1}^\infty (\mathsf{Y}_j,\mathsf{X}_j)$ be as in Definition countablesum. Then:

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • ...and 33 more