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Dynamical Systems and Commutants in Non-Commutative Algebras
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0001-9658-1222
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis work is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. In Mathematics, it is well known that matrix multiplication (or composition of linear operators on a finite dimensional vector space) is not always commutative. Commuting matrices or more general linear or non-linear operators play an essential role in Mathematics and its applications in Physics and Engineering. Many important relations in Mathematics, Physics and Engineering are represented by operators satisfying a number of commutation relations. Such commutation relations are key in areas such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others.

In Chapter 2 of this thesis we treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain onedimensional dynamical systems.

In Chapter 3, we treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra.

In Chapters 4 and 5, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non-decreasing sequence of algebras.

In Chapter 6 we give a description of the centralizer of the coefficient algebra in the Ore extension of the algebra of functions on a countable set with finite support.

Place, publisher, year, edition, pages
Västerås: Mälardalen University , 2018. , p. 140
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 258
National Category
Natural Sciences
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-39000ISBN: 978-91-7485-381-0 (print)OAI: oai:DiVA.org:mdh-39000DiVA, id: diva2:1197766
Public defence
2018-05-29, Kappa, Mälardalens högskola, Västerås, 13:00 (English)
Opponent
Supervisors
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2018-04-17 Created: 2018-04-13 Last updated: 2018-05-07Bibliographically approved
List of papers
1. On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems
Open this publication in new window or tab >>On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems
2014 (English)In: 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014 Conference date: 15–18 July 2014 Location: Narvik, Norway ISBN: 978-0-7354-1276-7 Editor: Seenith Sivasundaram Volume number: 1637 Published: 10 december 2014 / [ed] Seenith Sivasundaram, American Institute of Physics (AIP), 2014, p. 1110-1119Conference paper, Published paper (Refereed)
Abstract [en]

T. Persson and S. D. Sivestrov investigated representations of operators satisfying the relation XX* = F(X*X) in connection with periodic points and orbits of the map F. In particular they derived commutativity conditions for two monomials in operators A and B on a Hilbert space satisfying the relation AB = BF(A). In this article we shall apply their results to special one-dimensional dynamical systems and and give an explicit description of the interplay between periodic orbits of one-dimensional piecewise polynomial maps and commutativity of monomials for special operators A and B. Furthermore we shall apply our results to derive conditions on β for the special case when F β is the β–shift dynamical system.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2014
National Category
Mathematics Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-27253 (URN)10.1063/1.4904686 (DOI)000347812200129 ()2-s2.0-85031861299 (Scopus ID)
Conference
10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014 Conference date: 15–18 July 2014 Location: Narvik, Norway
Available from: 2014-12-31 Created: 2014-12-31 Last updated: 2018-04-17Bibliographically approved
2. Crossed Product Algebras for Piece-Wise Constant Functions
Open this publication in new window or tab >>Crossed Product Algebras for Piece-Wise Constant Functions
2016 (English)In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rančić, Springer, 2016Chapter in book (Refereed)
Abstract [en]

In this paper we consider algebras of functions that are constant on the sets of a partition. We describe the crossed product algebras of the mentioned algebras with Z. We show that the function algebra is isomorphic to the algebra of all functions on some set. We also describe the commutant of the function algebra and finish by giving an example of piece-wise constant functions on a real line.

Place, publisher, year, edition, pages
Springer, 2016
Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 179
Keyword
crossed product algebras; function algebra; commutant; piece-wise constant functions
National Category
Mathematics Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33242 (URN)10.1007/978-3-319-42105-6_6 (DOI)2-s2.0-85012918640 (Scopus ID)978-3-319-42104-9 (ISBN)978-3-319-42105-6 (ISBN)
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-09-25 Created: 2016-09-25 Last updated: 2018-04-17Bibliographically approved
3. Commutants in Crossed Product Algebras for Piece-Wise Constant Functions
Open this publication in new window or tab >>Commutants in Crossed Product Algebras for Piece-Wise Constant Functions
2016 (English)In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and OptimizationEditors / [ed] Sergei Silvestrov; Milica Rančić, Springer, 2016, p. 95-108Chapter in book (Refereed)
Abstract [en]

In this paper we consider crossed product algebras of algebras of piecewiseconstant functions on the real line with Z. For an increasing sequence of algebras (in which case the commutants form a decreasing sequence), we describe the set difference between the corresponding commutants.

Place, publisher, year, edition, pages
Springer, 2016
Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 179
Keyword
crossed product algebras, piecewiseconstant functions, commutants
National Category
Mathematics Algebra and Logic
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33243 (URN)10.1007/978-3-319-42105-6_7 (DOI)2-s2.0-85013026914 (Scopus ID)
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-09-25 Created: 2016-09-25 Last updated: 2018-04-17Bibliographically approved
4. Commutants in crossed products for algebras of piecewise constant functions on the real line
Open this publication in new window or tab >>Commutants in crossed products for algebras of piecewise constant functions on the real line
(English)In: Axioms, ISSN 2075-1680, , p. 140Article in journal (Refereed) Submitted
Place, publisher, year, edition, pages
MDPI. p. 140
National Category
Natural Sciences
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-38997 (URN)
Available from: 2018-04-13 Created: 2018-04-13 Last updated: 2018-04-17Bibliographically approved
5. Ore extensions of function algebras
Open this publication in new window or tab >>Ore extensions of function algebras
2018 (English)Conference paper, Oral presentation with published abstract (Refereed)
National Category
Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-38999 (URN)
Conference
International Conference "Stochastic Processes and Algebraic Structures" (SPAS2017)
Available from: 2018-04-13 Created: 2018-04-13 Last updated: 2018-04-27Bibliographically approved

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