Vol. 263, No. 2, 2013

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Real closed separation theorems and applications to group algebras

Tim Netzer and Andreas Thom

Vol. 263 (2013), No. 2, 435–452
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 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 1-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
Authors
Tim Netzer
Mathematisches Institut
University of Leipzig
PF 10 09 20
D-04009 Leipzig
Germany
Andreas Thom
Mathematisches Institut
University of Leipzig
PF 10 09 20
D-04009 Leipzig
Germany