Tracial states on the \(\theta\)-deformed plane
type: Item
instance of: scholarly article
title: Tracial states on the \(\theta\)-deformed plane
zbMATH Open document ID: 1196.58003
author: Ken Miyake
published in: Tokyo Journal of Mathematics
publication date: 2010-07-09T00:00:00Z
review text: A tracial state \(\phi:{\mathcal A}\to\mathbb{C}\), \({\mathcal A}\) is a unital \(*\)-algebra (in this paper, \(^-\) is used instead of \(*\)) such that \[ \phi(\overline aA)\geq 0,\;\phi(\mathbf{1}_A)= 1,\;\phi(xy)= \phi(yx), \] where \(\mathbf{1}_A\) is the unit element of \({\mathcal A}\). The set of tracial states forms a convex set which is called the tracial state space. In this paper, tracial states on the \(\theta\)-deformed \(2m\)-plane \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) are constructed and the extremal points of the tracial state space of \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) are obtained. Here, \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) is the unital \(*\)-algebra generated by \(m\) elements \(z^i\), \(i= 1,\ldots,m\), with the commutation relation \[ z^i z^j= \lambda^{ij} z^j z^i,\;\overline z^i\overline z^j= \lambda^{ij}\overline z^j\overline z^i,\;z^i\overline z^j= \lambda^{ji}\overline z^j z^i\;(1\leq i,j\leq m), \] where \(\lambda^{ij}= e^{2\pi i\theta_{ij}}\) and \(\theta= (\theta_{ij})\) is an anti-symmetric real-valued matrix of degree \(m\). To construct tracial states, a function \(\phi: C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\to\mathbb C\) is introduced by \(\phi(z^i z^j)= \phi(z^j\overline z^i)= \lambda^{ij}\), \(\phi(z^i\overline z^j)= \phi(\overline z^j\overline z^i)= \lambda^{ji}\), \(\phi(z^j z^i)= \phi(\overline z^j z^i)= \phi(\overline z^i\overline z^j)= \phi(\overline z^i z^j)= 1\), \(\phi(z^i)= \phi(\overline z^i)= 0\), \(\phi(\mathbf{1})= 1\). As for higher-order monomials, putting \(z^1= w_1\), \(\overline z^1= w_2,\ldots, z^m= w_{2m-1}\), \(\overline z^m= w_{2m}\), \[ \phi(w_{i(1)}\cdots w_{i(n)})= \prod^n_{k< l} \phi(w_{i(k)} w_{i(l)}),\;i(1),\ldots,i(n)\in\mathbb{N},\;1\leq i(1),\ldots,i(n)\leq2m, \] is required. Let \(T^{i(1)\cdots i(t)}\) be the set of monomials formed by the generators \(\mathbf{1}\), \(z^{(i(1)},\overline z^{i(1)}, \ldots, z^{i(t)}\), \(\overline z^{i(t)}\). If there exists a monomial \(Y\in T^{i(…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 53D55
zbMATH DE Number: 5734880
zbMATH Keywords: tracial state
zbMATH Keywords: \(\theta\)-deformed \(2m\)-plane
zbMATH Keywords: tractal state space
zbMATH Keywords: extremal points of the tracial state space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1264170248
OpenAlex ID: W2073365698
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DOI: 10.3836/TJM/1264170248
description: scientific article; zbMATH DE number 5734880