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Monic representations of finite higher-rank graphs

Part of: Selfadjoint operator algebras Topological (rings and) algebras with an involution

Published online by Cambridge University Press:  06 September 2018

CARLA FARSI ,
ELIZABETH GILLASPY ,
PALLE JORGENSEN ,
SOORAN KANG  and
JUDITH PACKER
CARLA FARSI
Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395, USA email carla.farsi@colorado.edu, packer@euclid.colorado.edu
ELIZABETH GILLASPY
Affiliation:
Department of Mathematics, University of Montana, 32 Campus Drive #0864, Missoula, MT 59812-0864, USA email elizabeth.gillaspy@mso.umt.edu
PALLE JORGENSEN
Affiliation:
Department of Mathematics, 14 MLH, University of Iowa, Iowa City, IA 52242-1419, USA email palle-jorgensen@uiowa.edu
SOORAN KANG
Affiliation:
College of General Education, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, Republic of Korea email sooran09@cau.ac.kr
JUDITH PACKER
Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395, USA email carla.farsi@colorado.edu, packer@euclid.colorado.edu
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Abstract

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.

Keywords

monic representations𝛬-semibranching function systemsMarkov measuresprojective systems

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems 46K10: Representations of topological algebras with involution
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1238 - 1267
Copyright
© Cambridge University Press, 2018

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