Real Closed Separation Theorems and Applications to Group Algebras

TL;DR

This paper extends separation theorems to real closed fields, applies them to *-algebras and group algebras, and establishes new Positivstellensatz results with implications for group cohomology and property (T).

Contribution

It introduces a strong Hahn-Banach theorem over real closed fields and applies it to derive novel Positivstellensatz results for *-algebras and group algebras.

Findings

Separation of convex sets is possible over real closed extension fields.

The cone of sums of squares has an interior point iff the first cohomology vanishes.

For groups with property (T), the interior point result strengthens in the $\

Abstract

In this paper we prove a strong Hahn-Banach theorem: separation of disjoint convex sets by linear forms is possible without any further conditions, if the target field $\R$ is replaced by a more general real closed extension field. From this we deduce a general Positivstellensatz for *-algebras, involving representations over real closed fields. We investigate the class of group algebras in more detail. We show that the cone of sums of squares in the augmentation ideal has an interior point if and only if the first cohomology vanishes. For groups with Kazhdan's property (T) the result can be strengthened to interior points in the $\ell^1$-metric. We finally reprove some strong Positivstellens\"atze by Helton and Schm\"udgen, using our separation method.