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 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 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 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 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 Equilibrium states on higher-rank Toeplitz non-commutative solenoids

2019 ◽  
Vol 40 (11) ◽  
pp. 2881-2912 ◽  
Author(s):  
ZAHRA AFSAR ◽  
ASTRID AN HUEF ◽  
IAIN RAEBURN ◽  
AIDAN SIMS
Keyword(s):  
Continuous Functions ◽  
Equilibrium States ◽  
Probability Measures ◽  
Inverse Limits ◽  
Inverse Temperature ◽  
Higher Dimensional ◽  
Direct Limits ◽  
Higher Rank ◽  
Natural Dynamics

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.

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

scholarly journals Projective multi-resolution analyses arising from direct limits of Hilbert modules

2007 ◽  
Vol 100 (2) ◽  
pp. 317 ◽  
Author(s):  
Nadia S. Larsen ◽  
Iain Raeburn
Keyword(s):  
Directed Graph ◽  
Hilbert Spaces ◽  
Path Space ◽  
Hilbert Modules ◽  
C Algebras ◽  
Infinite Path ◽  
Direct Limits ◽  
New Applications ◽  
Shed Light

The authors have recently shown how direct limits of Hilbert spaces can be used to construct multi-resolution analyses and wavelets in $L^2(\mathsf R)$. Here they investigate similar constructions in the context of Hilbert modules over $C^*$-algebras. For modules over $C(\mathsf T^n)$, the results shed light on work of Packer and Rieffel on projective multi-resolution analyses for specific Hilbert $C(\mathsf T^n)$-modules of functions on $\mathsf R^n$. There are also new applications to modules over $C(C)$ when $C$ is the infinite path space of a directed graph.

scholarly journals KMS States on the Operator Algebras of Reducible Higher-Rank Graphs

2017 ◽  
Vol 88 (1) ◽  
pp. 91-126 ◽  
Author(s):  
Astrid an Huef ◽  
Sooran Kang ◽  
Iain Raeburn
Keyword(s):  
Operator Algebras ◽  
Kms States ◽  
Higher Rank

scholarly journals Remarks on some fundamental results about higher-rank graphs and their C*-algebras

2013 ◽  
Vol 56 (2) ◽  
pp. 575-597 ◽  
Author(s):  
Robert Hazlewood ◽  
Iain Raeburn ◽  
Aidan Sims ◽  
Samuel B. G. Webster
Keyword(s):  
Equivalence Relation ◽  
Direct Proof ◽  
Explicit Description ◽  
C Algebras ◽  
Infinite Path ◽  
Higher Rank ◽  
Commuting Squares

AbstractResults of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

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