scholarly journals KMS states on the C⁎-algebra of a higher-rank graph and periodicity in the path space

2015 ◽  
Vol 268 (7) ◽  
pp. 1840-1875 ◽  
Author(s):  
Astrid an Huef ◽  
Marcelo Laca ◽  
Iain Raeburn ◽  
Aidan Sims
Keyword(s):  
Path Space ◽  
Kms States ◽  
C Algebra ◽  
Higher Rank

scholarly journals Monic representations of finite higher-rank graphs

2018 ◽  
Vol 40 (5) ◽  
pp. 1238-1267 ◽  
Author(s):  
CARLA FARSI ◽  
ELIZABETH GILLASPY ◽  
PALLE JORGENSEN ◽  
SOORAN KANG ◽  
JUDITH PACKER
Keyword(s):  
Continuous Functions ◽  
Path Space ◽  
Cyclic Vector ◽  
Kms States ◽  
Infinite Path ◽  
Representation Model ◽  
Higher Rank ◽  
Universal Representation ◽  
Comprehensive Study

In this paper, we define the notion of monic representation for the$C^{\ast }$-algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative$C^{\ast }$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the$\unicode[STIX]{x1D6EC}$-semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs.J. Math. Anal. Appl. 434 (2015), 241–270) and also provide a universal representation model for non-negative monic representations.

scholarly journals Spectral triples for higher-rank graph $C^*$-algebras

2020 ◽  
Vol 126 (2) ◽  
pp. 321-338
Author(s):  
Carla Farsi ◽  
Elizabeth Gillaspy ◽  
Antoine Julien ◽  
Sooran Kang ◽  
Judith Packer
Keyword(s):  
Brownian Motion ◽  
Dirac Operator ◽  
Wavelet Decomposition ◽  
Path Space ◽  
Spectral Triple ◽  
Spectral Triples ◽  
C Algebra ◽  
C Algebras ◽  
Strongly Connected ◽  
Higher Rank

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(\Lambda )$ of a strongly connected finite higher-rank graph Λ. Our spectral triple builds on an approach used by Consani and Marcolli to construct spectral triples for Cuntz-Krieger algebras. We prove that our spectral triples are intimately connected to the wavelet decomposition of the infinite path space of Λ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of our spectral triple. The paper concludes by discussing other properties of the spectral triple, namely, θ-summability and Brownian motion.

scholarly journals Spatial realisations of KMS states on the C⁎-algebras of higher-rank graphs

2015 ◽  
Vol 427 (2) ◽  
pp. 977-1003 ◽  
Author(s):  
Astrid an Huef ◽  
Sooran Kang ◽  
Iain Raeburn
Keyword(s):  
Kms States ◽  
C Algebras ◽  
Higher Rank

scholarly journals Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

10.53733/90 ◽  
2021 ◽  
Vol 52 ◽  
pp. 109-143
Author(s):  
Astrid An Huef ◽  
Marcelo Laca ◽  
Iain Raeburn
Keyword(s):  
Regular Representation ◽  
Crossed Product ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Natural Numbers ◽  
Kms States ◽  
C Algebra ◽  
Affine Semigroup ◽  
Natural Dynamics ◽  
The Right

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

scholarly journals Cocycles on groupoids arising from -actions

2021 ◽  
pp. 1-32
Author(s):  
CARLA FARSI ◽  
LEONARD HUANG ◽  
ALEX KUMJIAN ◽  
JUDITH PACKER
Keyword(s):  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Compact Metric Space ◽  
Kms States ◽  
Existence And Uniqueness Results ◽  
Commuting Operators ◽  
Uniqueness Results ◽  
Unit Space ◽  
Higher Rank

Abstract We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

scholarly journals Separable representations, KMS states, and wavelets for higher-rank graphs

2016 ◽  
Vol 434 (1) ◽  
pp. 241-270 ◽  
Author(s):  
Carla Farsi ◽  
Elizabeth Gillaspy ◽  
Sooran Kang ◽  
Judith A. Packer
Keyword(s):  
Kms States ◽  
Higher Rank

scholarly journals RECOVERING THE BOUNDARY PATH SPACE OF A TOPOLOGICAL GRAPH USING POINTLESS TOPOLOGY

2020 ◽  
pp. 1-17
Author(s):  
GILLES G. DE CASTRO
Keyword(s):  
Path Space ◽  
Topological Graph ◽  
Topological Graphs ◽  
Locally Compact ◽  
C Algebra ◽  
Compact Spaces ◽  
Compact Topology ◽  
Boundary Path ◽  
Definition Of ◽  
Locally Compact Spaces

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

scholarly journals KMS states onC⁎-algebras associated to higher-rank graphs

2014 ◽  
Vol 266 (1) ◽  
pp. 265-283 ◽  
Author(s):  
Astrid an Huef ◽  
Marcelo Laca ◽  
Iain Raeburn ◽  
Aidan Sims
Keyword(s):  
Kms States ◽  
Higher Rank
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