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The socle of subshift algebras, with applications to subshift conjugacy

Part of: Topological dynamics Rings and algebras arising under various constructions

Published online by Cambridge University Press:  18 November 2024

Daniel Gonçalves Open the ORCID record for Daniel Gonçalves [Opens in a new window]  and
Danilo Royer
Daniel Gonçalves
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil ([email protected]) (corresponding author)
Danilo Royer
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil ([email protected])
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Abstract

We introduce the concept of ‘irrational paths’ for a given subshift and useit to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott–Tomforde–Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.

Keywords

conjugacyLeavitt path algebrasminimal left idealsOTW-subshiftssoclesubshift algebras

MSC classification

Primary: 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
Secondary: 37B10: Symbolic dynamics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 26
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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