On the K-Theory of Graph C
              *-Algebras

Cornelissen, Gunther; Lorscheid, Oliver; Marcolli, Matilde

doi:10.1007/s10440-008-9208-4

Cited by 10 publications

(28 citation statements)

References 21 publications

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“…for some polynomial D + (u) of degree 2|V | − 1 with D + (0) = 0. Plugging this into the generating series (8) and take logs, we find…”

Section: Second Proof Of Theorem A(i) the Results Of Bassmentioning

confidence: 99%

Edge Reconstruction of the Ihara Zeta Function

Cornelissen

Kool²

2018

Electron. J. Combin.

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We show that if a graph G has average degree d ≥ 4, then the Ihara zeta function of G is edgereconstructible. We prove some general spectral properties of the edge adjacency operator T : it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if d > 4, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of T (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once.The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.Date: August 22, 2018 (version 2.0). 2010 Mathematics Subject Classification. 05C50, 05C38, 11M36, 37F35, 53C24.A degree-one vertex is called an end-vertex. All results in this paper hold for connected finite undirected multigraphs without end-vertices, and from now on we will use the word "graph" for such multigraphs.If e = {v 1 , v 2 } ∈ E, we denote by e = (v 1 , v 2 ) the edge e with a chosen orientation, and by e = (v 2 , v 1 ) the same edge with the inverse orientation to that of e. Let o( e) = v 1 denote the origin of e and t( e) = v 2 its end point. If there are multiple edges between v 1 , v 2 then we will label them e i = (v 1 , v 2 ) i . A nonbacktracking edge walk of length n is a sequence e 1 e 2 ....e n of edges such that t(e i ) = o(e i+1 ), but e i+1 = e i . We call it tailless if e n = e 1 . Just like walks in the graph can be studied using the adjacency matrix, nonbacktracking walks are captured by the edge adjacency matrix T = T G studied by Sunada [24], Hashimoto [14] and Bass [2]. Letting E denote the set of oriented edges of G for any possible choice of orientation, so | E | = 2|E|, T is defined to be the 2|E| × 2|E| matrix, in which the rows and columns are indexed by E, and T e 1 , e 2 = 1 if t( e 1 ) = o( e 2 ) but e 2 = e 1 ; 0 otherwise.

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“…for some polynomial D + (u) of degree 2|V | − 1 with D + (0) = 0. Plugging this into the generating series (8) and take logs, we find…”

Section: Second Proof Of Theorem A(i) the Results Of Bassmentioning

confidence: 99%

Edge Reconstruction of the Ihara Zeta Function

Cornelissen

Kool²

2018

Electron. J. Combin.

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“…It is also strictly isomorphic as C * -algebra to the CuntzKrieger algebra of the edge adjacency matrix (=Bass-Hashimoto operator, cf. [9]) of a graph with one vertex and g X loops, i.e., the Cuntz-Krieger algebra corresponding to the matrix…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…The situation becomes much simpler if one instead studies (stable) isomorphism of the Cuntz-Krieger algebras associated to the edge adjancency matrix (Bass-Hashimoto operator) of the graph. This operator T X is defined as follows (compare with [1] or [9]). Choose a random orientation of the edged {e 1 , .…”

Section: Remarkmentioning

confidence: 99%

“…This operator is very natural for counting non-backtracking paths in the graph (relating to the Ihara zeta function). In [9], one finds a geometric study of this classification problem, and the answer is that stable isomorphism of such algebras for graphs X and Y is the same as equality of Betti numbers g X = g Y , and strict isomorphism means that we also have and equality…”

Section: Remarkmentioning

confidence: 99%

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Graph reconstruction and quantum statistical mechanics

Cornelissen¹,

Marcolli²

2013

Journal of Geometry and Physics

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Abstract. We study in how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.

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“…It seems that in general we must understand algebras that admit There are two motivations for this study. First there is the observation [5] that there is a way to associate directed graphs to Mumford curves and that from the corresponding graph C * -algebras we might extract topological information about the curves. It eventuates that SU q (2) is an example of a graph algebra that shares with the Mumford curve graph algebras the property that it does not admit faithful traces but does admit faithful KMS states for non-trivial circle actions.…”

Section: Introductionmentioning

confidence: 99%

Spectral flow invariants and twisted cyclic theory for the Haar state on SUq(2)

Carey

Rennie

Tong

2009

Journal of Geometry and Physics

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a b s t r a c tIn [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K -theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU q (2) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial 'eta' contributions to this index.

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On the K-Theory of Graph C *-Algebras

Cited by 10 publications

References 21 publications

Edge Reconstruction of the Ihara Zeta Function

Edge Reconstruction of the Ihara Zeta Function

Graph reconstruction and quantum statistical mechanics

Spectral flow invariants and twisted cyclic theory for the Haar state on SUq(2)

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