Operator systems and positive extensions over discrete groups

TL;DR

This paper explores the extension problem for positive semi-definite functions on discrete groups using operator system theory, linking it to sums of squares and noncommutative geometry.

Contribution

It provides an operator system framework that characterizes extension and factorization properties, connecting them to duality, nuclearity, and C*-envelopes.

Findings

Extension property characterized by a quotient map on Fourier--Stieltjes algebra

Factorization property linked to a complete order embedding into the group C*-algebra

Operator system techniques relate to spectral and Fourier truncations in noncommutative geometry

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.