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NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY OF KAZHDAN’S PROPERTY (T)

Part of: Selfadjoint operator algebras Special aspects of infinite or finite groups Locally compact groups and their algebras

Published online by Cambridge University Press:  30 July 2014

Narutaka Ozawa
Narutaka Ozawa*
Affiliation:
RIMS, Kyoto University, 606-8502, Japan ([email protected])
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Abstract

It is well known that a finitely generated group ${\rm\Gamma}$ has Kazhdan’s property (T) if and only if the Laplacian element ${\rm\Delta}$ in $\mathbb{R}[{\rm\Gamma}]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in $\mathbb{R}[{\rm\Gamma}]$. Namely, ${\rm\Gamma}$ has property (T) if and only if there exist a constant ${\it\kappa}>0$ and a finite sequence ${\it\xi}_{1},\ldots ,{\it\xi}_{n}$ in $\mathbb{R}[{\rm\Gamma}]$ such that ${\rm\Delta}^{2}-{\it\kappa}{\rm\Delta}=\sum _{i}{\it\xi}_{i}^{\ast }{\it\xi}_{i}$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in $\mathbb{R}[{\rm\Gamma}]$, possibly with the assistance of computers.

Keywords

Kazhdan’s property (T)Positivstellensätzspectral gap

MSC classification

Primary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 22D10: Unitary representations of locally compact groups 20F10: Word problems, other decision problems, connections with logic and automata
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 1 , January 2016 , pp. 85 - 90
Copyright
© Cambridge University Press 2014 

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