Gauge-invariant uniqueness theorems for \(P\)-graphs
type: Item
instance of: scholarly article
title: Gauge-invariant uniqueness theorems for \(P\)-graphs
author: Robert Huben
author: S. Kaliszewski
author: Nadia S. Larsen
author: John Quigg
published in: Mathematica Scandinavica
publication date: 2024-06-25T00:00:00Z
review text: ``Uniqueness theorem'' means in this paper an answer to the following question: given a \(C^*\)-algebra \(A\), when is a homomorphism \(\pi:A\to B\) faithful? The ``gauge-invariant'' phrase comes from an equivariance condition: if \(A\) carries an action \(\Gamma \curvearrowright A\) of a group \(\Gamma\), we can ask for the existence of an action \(\Gamma \curvearrowright B\). In this paper, a version of the gauge-invariant uniqueness theorem is proved using maximal co-actions in the context of discrete groups, since that maximal co-actions provide a natural starting point to state and prove such uniqueness theorems.\N\NAn \textit{ordered group} is a pair \((G,P)\), where \(G\) is a group and \(P\) a submonoid such that \(P\cap P^{-1}=\{e\}\). There is a partial order on \(G\) defined by \(a \leq b\) if \(a^{-1}b\in P\). A \textit{weakly quasi-lattice ordered} (WQLO) group is an ordered group \((G,P)\) such that for all \(p,\,q\in P\), if \(\{p,q\}\) is bounded, then it has a least upper bound in \(P\), denoted by \(p\vee q\). In this paper, \(P\) will always refer to a WQLO group \((G,P)\). A \(P\)-graph is a countable category \(\Lambda\) together a functor \(d:\Lambda\to P\) such that for all \(\alpha\in\Lambda\) and \(p,\,q\in P\) such that \(d(\alpha)=pq\), there exist unique \(\beta,\,\gamma\in\Lambda\) such that \(\alpha=\beta\gamma\) and \(d(\beta)=p\) (and hence \(d(\gamma)=q\)). The elements of \(\Lambda\) are called \textit{paths} and the objects are identified with the identity morphisms, called \textit{vertices}. The set of all vertices is denoted by \(\Lambda^0\). We say that \(\Lambda\) is \textit{finitely aligned} if \(\bigvee S=\{\mu\in\bigcap\limits_{\gamma\in S}\gamma \Lambda:d(\mu)=\vee d(S)\}\) is finite for all finite \(S\subset\Lambda\).\N\NA \textit{representation} of a finitely aligned \(P\)-graph \(\Lambda\) in a \(C^*\)-algebra \(B\) is a map \(t:\Lambda\to B\) such that for all \(\alpha,\,\beta\in \Lambda\) we have that \(t_\alpha t_\bet…
reviewed by: M. V. Ferrer
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 7870741
zbMATH Keywords: \(P\)-graph
zbMATH Keywords: representation
zbMATH Keywords: co-action
MaRDI profile type: Publication
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DOI: 10.7146/MATH.SCAND.A-139786
description: scientific article; zbMATH DE number 7870741