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Hilbert evolution algebras and its connection with discrete-time Markov chains

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Abstract

Evolution algebras are non-associative algebras. In this work we provide an extension of this class of algebras, in a framework of Hilbert spaces, and illustrate the applicability of our approach by discussing a connection with discrete-time Markov chains with infinite countable state space. Specifically, if we associate to each possible state of such a Markov process a generator of the Hilbert evolution algebra structure, then the whole dynamics of the process can be described through consecutive applications of the evolution operator, provided certain boundedness conditions on the transition probabilities hold.

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Acknowledgements

Part of this work was carried out during a visit of P.C. at the Universidade Federal de Pernambuco (UFPE); and visits of P.M.R. at the Universidade Federal do ABC (UFABC) and at the Universidad Nacional de la Patagonia “San Juan Bosco” (UNPSJB). The visit at UNPSJB was during the realization of the School EMALCA 2019. The authors are grateful with these institutions, and with the organizers of the School, for their hospitality and support. Part of this work has been supported by Fundação de Amparo á Pesquisa do Estado de São Paulo-FAPESP (Grant 2017/10555-0).

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Author notes

  1. Sebastian J. Vidal, Paula Cadavid and Pablo M. Rodriguez have contributed equally to this work.

Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ingeniería, Universidad Nacional de la Patagonia “San Juan Bosco”, Km 4, 9000, Comodoro Rivadavia, Chubut, Argentina

    Sebastian J. Vidal

  2. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Avenida dos Estados 5001, Santo André, SP, 09210-580, Brazil

    Paula Cadavid

  3. Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco, Av. Jornalista Aníbal Fernandes, 497-629, Cidade Universitária, Recife, PE, 50740-540, Brazil

    Pablo M. Rodriguez

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  1. Sebastian J. Vidal
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  2. Paula Cadavid
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  3. Pablo M. Rodriguez
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Corresponding author

Correspondence to Pablo M. Rodriguez.

Additional information

Communicated by Rahul Roy.

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Vidal, S.J., Cadavid, P. & Rodriguez, P.M. Hilbert evolution algebras and its connection with discrete-time Markov chains. Indian J Pure Appl Math 54, 883–894 (2023). https://doi.org/10.1007/s13226-022-00304-y

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