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Quadratic subproduct systems, free products, and their C*-algebras

Francesca Arici, Yufan Ge

Abstract

Motivated by the interplay between quadratic algebras, noncommutative geometry, and operator theory, we introduce the notion of quadratic subproduct systems of Hilbert spaces. Specifically, we study the subproduct systems induced by a finite number of complex quadratic polynomials in noncommuting variables, and describe their Toeplitz and Cuntz--Pimsner algebras. Inspired by the theory of graded associative algebras, we define a free product operation in the category of subproduct systems and show that this corresponds to the reduced free product of the Toeplitz algebras. Finally, we obtain results about the K-theory of the Toeplitz and Cuntz--Pimsner algebras of a large class of quadratic subproduct systems.

Quadratic subproduct systems, free products, and their C*-algebras

Abstract

Motivated by the interplay between quadratic algebras, noncommutative geometry, and operator theory, we introduce the notion of quadratic subproduct systems of Hilbert spaces. Specifically, we study the subproduct systems induced by a finite number of complex quadratic polynomials in noncommuting variables, and describe their Toeplitz and Cuntz--Pimsner algebras. Inspired by the theory of graded associative algebras, we define a free product operation in the category of subproduct systems and show that this corresponds to the reduced free product of the Toeplitz algebras. Finally, we obtain results about the K-theory of the Toeplitz and Cuntz--Pimsner algebras of a large class of quadratic subproduct systems.

Paper Structure

This paper contains 21 sections, 42 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on subproduct systems and their algebras
  3. Toeplitz and Cuntz--Pimsner algebras of subproduct systems
  4. Quadratic subproduct systems from quadratic algebras
  5. Quadratic Algebras and Their Hilbert Series
  6. Free products of quadratic algebras and their Hilbert series
  7. Quadratic subproduct systems
  8. The importance of being generic
  9. Free products of subproduct systems and their Fock spaces
  10. Fusion Rules for free product subproduct systems
  11. Fock spaces of free products are free products of Fock spaces
  12. Toeplitz algebras and KK-theory
  13. Free products of Toeplitz algebras and functoriality
  14. KK-theory and nuclearity for free products
  15. A case study: free products of Temperley--Lieb subproduct systems
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathcal{H}$ and $\mathcal{K}$ be standard subproduct systems of Hilbert spaces. Assume that the Toeplitz algebras $\mathbb{T}_{\mathcal{H}}$ and $\mathbb{T}_{\mathcal{K}}$ are both nuclear and KK-equivalent to the complex numbers. Then so is the Toeplitz algebra $\mathbb{T}_{\mathcal{H}\star \

Theorems & Definitions (85)

  • Theorem
  • Proposition 2.1: ShSo09
  • Proposition 2.2: ShSo09
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3: cf. PoPo
  • Definition 3.4: cf. PoPo
  • Proposition 3.5: PoPo
  • Remark 3.6
  • Example 3.7
  • ...and 75 more