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Operator systems and positive extensions over discrete groups

Evgenios T. A. Kakariadis, Malte Leimbach, Ivan G. Todorov, Walter D. van Suijlekom

Abstract

The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin in the 1960s that this is closely related to the problem of finding sums of squares factorisations of positive elements in the group C*-algebra. We give an operator system perspective at these two problems explaining their equivalence: the extension property is characterised by a certain quotient map on the Fourier--Stieltjes algebra, and the factorisation property by a certain complete order embedding into the group C*-algebra. These properties are linked to the duality of the operator systems which have recently emerged from spectral and Fourier truncations in noncommutative geometry. We exemplify how one can relate certain extension problems to operator system techniques such as nuclearity and the C*-envelope.

Operator systems and positive extensions over discrete groups

Abstract

The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin in the 1960s that this is closely related to the problem of finding sums of squares factorisations of positive elements in the group C*-algebra. We give an operator system perspective at these two problems explaining their equivalence: the extension property is characterised by a certain quotient map on the Fourier--Stieltjes algebra, and the factorisation property by a certain complete order embedding into the group C*-algebra. These properties are linked to the duality of the operator systems which have recently emerged from spectral and Fourier truncations in noncommutative geometry. We exemplify how one can relate certain extension problems to operator system techniques such as nuclearity and the C*-envelope.

Paper Structure

This paper contains 24 sections, 26 theorems, 200 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Operator system preliminaries
  3. Order unit spaces
  4. Matrix order unit spaces and operator systems
  5. Archimedeanisation
  6. Quotients
  7. Tensor products
  8. $\mathrm{C}^*$-covers
  9. Coproducts and amalgamated direct sums
  10. Colimits
  11. Operator spaces
  12. Complete order embeddings and quotient maps
  13. Positive semi-definite functions and the Fourier operator system
  14. Sums of squares and Toeplitz matrices
  15. The sums of squares operator system and the Fejér--Riesz property
  16. ...and 9 more sections

Key Result

Lemma 2.1

The Archimedeanisation is a functor from $\mathbf{MOU}$ to $\mathbf{OSy}$.

Figures (11)

  • Figure 1: The operator systems (in the second line) and their respective matrix ordered vector space duals (in the first line), together with the canonical ucp maps discussed in this note; here $\Gamma$ is a discrete group, $\Delta \subseteq \Gamma$ a positivity domain and $\Sigma \subseteq \Gamma$ a finite subset with $\Sigma\Sigma^{-1} \subseteq \Delta$.
  • Figure 2: The universal property of the Archimedeanisation.
  • Figure 3: The universal properties of the maximal and minimal $\mathrm{C}^*$-cover.
  • Figure 4: The universal property of the amalgamated direct sum.
  • Figure 5: The universal property of the colimit.
  • ...and 6 more figures

Theorems & Definitions (79)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 69 more