The \(C^*\)-algebras of row-finite graphs
type: Item
instance of: scholarly article
title: The \(C^*\)-algebras of row-finite graphs
zbMATH Open document ID: 0976.46041
author: Teresa Bates
author: David Pask
author: Iain Raeburn
author: Wojciech Szymański
published in: The New York Journal of Mathematics
publication date: 2000-11-30T00:00:00Z
full work available at URL: https://eudml.org/doc/121447
full work available at URL: 6-14nf.htm
review text: Let \(E\) be a row-finite directed graph, i.e., each vertex emits at most finitely many edges. Denote the vertices by \(v\)'s, the edges by \(e\)'s and let \(s(e)\) and \(r(e)\) be the source and range, respectively, of the edge \(e\). A Cuntz-Krieger E-family in a C\(^{*}\)-algebra consists of partial isometries \(s_{e}\) and mutually orthogonal projections \(p_{v}\) are such that \(s^{*}_{e}s_{e} = p_{r(e)}\), and for a \(v\) which is not a sink, \( p_{v} = \sum_{e:s(e)=v}{s_{e}s^{*}_{e}}\). Let \(C^{*}(E)\) denote the universal C\(^{*}\)-algebra generated by these relations [cf. \textit{A. Kumjian, D. Pask} and \textit{I. Raeburn}, Pac. J. Math. 184, No. 1, 161-174 (1998; Zbl 0717.46056)]. If \(\{ s_{e},p_{v}\}\) is an E-family generating \(C^{*}(E)\) then, for \(z \in {\mathbf{T}}\), also \(\{ zs_{e},p_{v} \}\) is an E-family generating \(C^{*}(E)\). The universal property of \(C^{*}(E)\) gives an endomomorphism \(\gamma_{z}\) of \(C^{*}(E)\) such that \(\gamma_{z}\{ s_{e},p_{v}\} = \{ zs_{e}, p_{v} \}\), and \(\gamma\) is a strongly continuous action of the ``gauge group'' \({\mathbf{T}}\) on \(C^{*}(E)\). The authors prove the following gauge-invariant uniqueness theorem: suppose \(\{ S_{e},P_{v}\}\) is a concrete E-family, the \(P_{v}\) assumed all non-zero, and that \(\pi\) is the representation \(\{ s_{e},p_{v}\} \mapsto \{ S_{e},P_{v}\}\); if there is a strongly continuous action \(\beta\) of \({\mathbf{T}}\) on \(C^{*}(S_{e},P_{v})\) such that \(\pi.\gamma_{z} =\beta_{z}.\pi\) for \(z \in {\mathbf{T}}\) then \(\pi\) is faithful. This gauge-invariance eliminates the necessity, for existence of the Cuntz algebras, of the condition I set in [\textit{J. Cuntz} and \textit{W. Krieger}, Invent. Math. 56, 251-268 (1980; Zbl 0434.46045)]. The authors effectively eliminate any sinks which occur by adding an infinite tail to each sink. A finite graph with a sink is converted to an infinite graph without the sink. The graph \(E\) extends to a larger graph \(F\) and …
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 1531353
zbMATH Keywords: C\(^{*}\)-algebra
zbMATH Keywords: row-finite directed graph
zbMATH Keywords: sink
zbMATH Keywords: Cuntz-Krieger relations
zbMATH Keywords: primitive ideal space
zbMATH Keywords: gauge group
zbMATH Keywords: gauge-invariant uniqueness theorem
zbMATH Keywords: Cuntz algebras
zbMATH Keywords: maximal tail
reviewed by: Ilie Valuşescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1531353