Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations
In order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-05-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/5/532 |
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| author | Vitonofrio Crismale Francesco Fidaleo Maria Elena Griseta |
| author_facet | Vitonofrio Crismale Francesco Fidaleo Maria Elena Griseta |
| author_sort | Vitonofrio Crismale |
| collection | DOAJ |
| description | In order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>, that is that generated by left and right hand-side partial shifts, the monoid of all strictly increasing maps whose range has finite complement, and finally the collection of all strictly increasing maps of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>. We show that such three monoids are strictly ordered, and the second-named one is the semidirect product between the first and the action of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula> generated by the one-step shift. Even if the definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the last monoid on the indices, we see that spreadability can be directly stated in terms of invariance with respect to the action of the first monoid. Concerning the stochastic processes involving the concrete boolean <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>∗</mo> </msup> </semantics> </math> </inline-formula>-algebra generated by the annihilators acting on the boolean Fock space (i.e., the concrete <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>∗</mo> </msup> </semantics> </math> </inline-formula>-algebra satisfying the boolean commutation relations), we study their spreadability directly in terms of the invariance under the monoid generated by all strictly increasing maps whose range has finite complement because, for this case, such an investigation appears more direct and manageable. Finally, we present the version of the Ryll–Nardzewski theorem for the boolean case, establishing that spreadable, exchangeable and stationary stochastic processes coincide, and describing their common structure. |
| first_indexed | 2024-03-10T19:57:57Z |
| format | Article |
| id | doaj.art-ee8e5cb28e544fbdb448e98e58d1947a |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T19:57:57Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-ee8e5cb28e544fbdb448e98e58d1947a2023-11-19T23:50:13ZengMDPI AGEntropy1099-43002020-05-0122553210.3390/e22050532Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation RelationsVitonofrio Crismale0Francesco Fidaleo1Maria Elena Griseta2Dipartimento di Matematica, Università degli studi di Bari, Via E. Orabona, 4, 70125 Bari, ItalyDipartimento di Matematica, Università degli studi di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, ItalyDipartimento di Matematica, Università degli studi di Bari, Via E. Orabona, 4, 70125 Bari, ItalyIn order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>, that is that generated by left and right hand-side partial shifts, the monoid of all strictly increasing maps whose range has finite complement, and finally the collection of all strictly increasing maps of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>. We show that such three monoids are strictly ordered, and the second-named one is the semidirect product between the first and the action of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula> generated by the one-step shift. Even if the definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the last monoid on the indices, we see that spreadability can be directly stated in terms of invariance with respect to the action of the first monoid. Concerning the stochastic processes involving the concrete boolean <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>∗</mo> </msup> </semantics> </math> </inline-formula>-algebra generated by the annihilators acting on the boolean Fock space (i.e., the concrete <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>∗</mo> </msup> </semantics> </math> </inline-formula>-algebra satisfying the boolean commutation relations), we study their spreadability directly in terms of the invariance under the monoid generated by all strictly increasing maps whose range has finite complement because, for this case, such an investigation appears more direct and manageable. Finally, we present the version of the Ryll–Nardzewski theorem for the boolean case, establishing that spreadable, exchangeable and stationary stochastic processes coincide, and describing their common structure.https://www.mdpi.com/1099-4300/22/5/532noncommutative probabilitynoncommutative dynamical systemsspreadable stochastic processesstates |
| spellingShingle | Vitonofrio Crismale Francesco Fidaleo Maria Elena Griseta Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations Entropy noncommutative probability noncommutative dynamical systems spreadable stochastic processes states |
| title | Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations |
| title_full | Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations |
| title_fullStr | Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations |
| title_full_unstemmed | Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations |
| title_short | Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations |
| title_sort | spreadability for quantum stochastic processes with an application to boolean commutation relations |
| topic | noncommutative probability noncommutative dynamical systems spreadable stochastic processes states |
| url | https://www.mdpi.com/1099-4300/22/5/532 |
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