Real closed separation theorems and applications to group algebras

Tim Netzer and Andreas Thom

Vol. 263 (2013), No. 2, 435–452

DOI: 10.2140/pjm.2013.263.435

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 ℓ¹-metric. We finally reprove some strong Positivstellensätze by Helton and Schmüdgen, using our separation method.

Keywords

real closed fields, group rings, Kazhdan’s property (T), sums of squares

Mathematical Subject Classification 2010

Primary: 46H15

Milestones

Received: 24 January 2012

Revised: 8 August 2012

Accepted: 13 August 2012

Published: 31 May 2013