A tracial Nullstellensatz.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1319.16019
author: Igor Klep
author: Špela Špenko
published in: Bulletin of the London Mathematical Society
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.1538
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 6355606
zbMATH Keywords: tracial Nullstellensatz
zbMATH Keywords: free algebras
zbMATH Keywords: trace algebras
zbMATH Keywords: matrix algebras
MaRDI profile type: Publication
OpenAlex ID: W2152360103
arXiv ID: 1311.1538
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DOI: 10.1112/BLMS/BDU064
Recommended article: Tracial Nullstellensätze
title: A tracial Nullstellensatz.
review text: The authors prove an analog of Hilbert's Nullstellensatz for polynomials in the free associative algebra when evaluated on matrix algebras. Let \(F\) be an algebraically closed field of characteristic 0, and let \(M(F)=\bigcup_{n\geq 0}M_n(F)\). Denote by \(F\langle X\rangle\) the free associative algebra freely generated over \(F\) by the set of \(g\) variables \(x_1,\ldots,x_g\). The main theorem of the paper is the following. Let \(f_1,\ldots,f_r,f\in F\langle X\rangle\) be polynomials. Then the vanishing of the traces of \(f_1(A),\ldots,f_r(A)\), for every choice of \(n\) and for every \(g\)-tuple of \(n\times n\) matrices, implies that \(f(A)\) is a traceless matrix if and only if \(f\) is cyclically equivalent to a linear combination of \(f_1,\ldots,f_r\), or else some linear combination of the \(f_i\) is cyclically equivalent to a nonzero scalar in \(F\langle X\rangle\). Recall that two polynomials \(f\) and \(g\) are cyclically equivalent if \(f-g\) is a sum of commutators in \(F\langle X\rangle\).
reviewed by: Plamen Koshlukov
description: scientific article; zbMATH DE number 6355606