Dynamical Systems and Commutants in Non-Commutative Algebras

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(1)Mälardalen University Doctoral Dissertation 258 Mälardalen University Doctoral Dissertation XXX Alex Behakanira Tumwesigye ”DYNAMICAL SYSTEMS AND COMMUTANTS IN NON-COMMUTATIVE ALGEBRAS!”. ISBN 978-91-7485-381-0 ISSN 1651-4238. 2018. Address: P.O. Box 883, SE-721 23 Västerås. Sweden Address: P.O. Box 325, SE-631 05 Eskilstuna. Sweden E-mail: info@mdh.se Web: www.mdh.se. Dynamical Systems and Commutants in Non-Commutative Algebras Alex Behakanira Tumwesigye.

(2) Mälardalen University Press Dissertations No. 258. DYNAMICAL SYSTEMS AND COMMUTANTS IN NON-COMMUTATIVE ALGEBRAS. Alex Behakanira Tumwesigye 2018. School of Education, Culture and Communication.

(3) Copyright © Alex Behakanira Tumwesigye, 2018 ISBN 978-91-7485-381-0 ISSN 1651-4238 Printed by E-Print AB, Stockholm, Sweden.

(4) Mälardalen University Press Dissertations No. 258. DYNAMICAL SYSTEMS AND COMMUTANTS IN NON-COMMUTATIVE ALGEBRAS. Alex Behakanira Tumwesigye. Akademisk avhandling som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras tisdagen den 29 maj 2018, 13.00 i Kappa, Mälardalens högskola, Västerås. Fakultetsopponent: Docent Olga Liivapuu, Estonian University of Life Sciences. Akademin för utbildning, kultur och kommunikation.

(5) Abstract 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 one-dimensional 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.. ISBN 978-91-7485-381-0 ISSN 1651-4238.

(6) Acknowledgements First and foremost, I would like to thank my supervisor Professor Sergei Silvestrov who accepted to take me up as his first student under the International Science Programme (ISP). Thank you Sergei for introducing me to this area of research that I have come to love and for the wonderful discussions that we had. Thank you for your patient guidance, enthusiastic encouragement and useful critiques during the development of this work. I would like to express my great appreciation to my co-supervisor Dr. Johan Richter for his valuable and constructive suggestions during the numerous academic discussions we had. I learned a lot during these discussions that I will take with me to wherever I go. My grateful thanks are also extended to my other co-supervisor Dr. Linus Carlsson for the academic engagements we had. I thank my supervisor from Uganda, Dr. Vincent Ssembatya who also shares my passion for music, for all the academic and the not so academic discussions. I would like to express my deep gratitude to my family, my wonderful wife Evas Tukahirwa and my three children Samantha Ainembabazi, Shantale Aineamani and Shanitah Ainemukama for the love and support you have offered me. Thank for enduring all those months when papa has been away in Sweden pursuing PhD studies. It has been comforting to know that I could count on your support throughout all this time. I would also like to thank, in a special way, my parents. My dad, Mr Vincent Behakanira for making me the man I am today. I thank my Auntie Fausta who has been our mother since our mum passed away, for all the support. I thank all my brothers and sisters for the encouragement. I would like to express my very great appreciation to the Swedish international development cooperation agency (Sida), International Science Programme in Mathematics (IPMS) and the East African Universities Mathematics Programme (EAUMP) for the financial support. In a special way, 5.

(7) Dynamical Systems and Commutants in Non-Commutative Algebras I thank Pravina Gajjar and Leif Abrahamsson at ISP, Uppsala for always providing quick answers and ensuring a comfortable stay in Sweden. Special thanks also go to the coordinators of EAUMP Makerere node, Profs. J. M. Mango , J. Kasozi and V. A. Ssembatya I would like to thank the staff at the School of education, culture and communication, Mälardalens University for providing a wonderful academic and research environment in Mathematics and Applied Mathematics. I am particularly greatful to Kristina Konpan who was always ready to attend to our administrative needs. Special thanks to various people who have in one way or the other made my stay in Sweden quite memorable. Farid Monsefi, Jan Skvaril, Milica Rancic, Maria Larsson, Anna Helfridsson and Marie Bergman, fellow PhD students under ISP, (my godchildren) Betuel Jesus Canhanga, Carole Ogutu, Jean-Paul Murara, (my godgrandchildren), Asaph Muhumuza, Pitos Biganda, Tin Nwe Aye, Samya Suleiman, Benard Abola, Elvice Ongonga, John Musonda and all other PhD students in Mathematics and Applied Mathematics, Mälardalens university for the nice moments we shared. Finally, I thank my colleagues in the Department of Mathematics, Makerere University for being such a wonderful family. Väster˚ as, May, 2018 Alex Behakanira Tumwesigye. 6.

(8) List of Papers The chapters 2 − 6 in this thesis are based, respectively, on the following papers: Paper A. Silvestrov, S. D., Tumwesigye, A. B., (2014), On monomial commutativity of operators satisfying commutation relations and periodic points for onedimensional dynamical systems, AIP Conference Proceedings, 1637, 11101119. Paper B. Richter, J., Silvestrov, S. D., Ssembatya, V. A., Tumwesigye, A. B., (2016). Crossed product algebras for piecewise constant functions, in Silvestrov S., Ranˇcić M., (eds.), Engineering Mathematics II. Algebraic, stochastic and analysis structures for networks, data classification and optimization, Springer Proceedings in Mathematics and Statistics, Springer, 75-93. Paper C. Richter, J., Silvestrov, S. D., Tumwesigye, A. B., (2016). Commutants in crossed products for algebras of piecewise constant functions, in Silvestrov S., Ranˇcić M., (eds.), Engineering Mathematics II. Algebraic, stochastic and analysis structures for networks, data classification and optimization, Springer Proceedings in Mathematics and Statistics, Springer, 95-108. Paper D. Richter, J., Silvestrov, S. D., Tumwesigye, A. B., (2017). Commutants in crossed products for algebras of piecewise constant functions on the real line, submitted to a journal. Paper E. Richter, J., Silvestrov, S. D., Tumwesigye, A. B., (2017). Ore extensions of function algebras, to appear in Proceedings of the International Conference ”Stochastic Processes and Algebraic Structures – From Theory Towards Applications” (SPAS 2017), Väster˚ as –Stockholm.. 7.

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(10) Contents. 1 Introduction. 15. 1.1 Definitions and general notions . . . . . . . . . . . . . . . . . . . .. 15. 1.2 Commuting elements in operator algebras associated to dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 1.2.1. An operator relation connected with dynamical systems . .. 1.2.2. Commutativity of monomials of operators on a finite-dimensional space and periodic orbits for one-dimensional dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 1.3 Crossed product algebras and C ∗ −crossed products . . . . . . . . . 1.3.1. 18. 22. Automorphisms induced by bijections . . . . . . . . . . . .. 26. 1.4 Ore extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 1.4.1. Centralizers in Ore extensions . . . . . . . . . . . . . . . . .. 29. 1.5 Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .. 30. 1.5.1. On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems . . . . . . . . . . . .. 30. 1.5.2. Crossed product algebras for piece-wise constant functions .. 30. 1.5.3. Commutants in crossed products for algebras of piece-wise constant functions . . . . . . . . . . . . . . . . . . . . . . .. 31. Commutants in crossed product algebras for piece-wise constant functions on the real line . . . . . . . . . . . . . . . .. 31. 1.5.4. 9.

(11) Dynamical Systems and Commutants in Non-Commutative Algebras 1.5.5. Ore extensions of function algebras . . . . . . . . . . . . . .. 31. 2 On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems 37 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 2.2 Monomial commutativity condition on F . . . . . . . . . . . . . . .. 39. 2.3 Operator monomial commutativity and periodic points of F . . . .. 41. 2.3.1. Monomial commutativity for the case when F is the β−shift 48. 3 Crossed product algebras for piece-wise constant functions 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55. 3.2 Definitions and a preliminary result . . . . . . . . . . . . . . . . .. 56. 3.3 Algebra of piece-wise constant functions . . . . . . . . . . . . . . .. 58. 3.3.1. Maximal Commutative Subalgebra . . . . . . . . . . . . . .. 63. 3.4 Algebra of piece-wise constant functions on the real line with N fixed jump points . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65. 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. 3.5.1. 3.5.2. Piece-wise constant functions with one jump point . . . . .. 69. 3.5.1.1. σ(I0 ) = I0 . . . . . . . . . . . . . . . . . . . . . .. 69. 3.5.1.2. σ(I0 ) = I1 . . . . . . . . . . . . . . . . . . . . . .. 70. Piece-wise constant functions with two jump points . . . .. 70. 3.5.2.1. σ(Iα ) = Iα for all α = 0, · · · , 4 . . . . . . . . . . .. 71. 3.5.2.2. σ(I0 ) = I1 ,. 3.5.2.3. σ(I0 ) = I1 , σ(I1 ) = I2 , σ(I2 ) = I0 and σ(Iα ) = Iα , α = 3, 4. . . . . . . . . . . . . . . . . . . . . .. 73. σ(I0 ) = I1 , σ(I1 ) = I2 , σ(I2 ) = I0 and σ(I3 ) = I4 , σ(I4 ) = I3 . . . . . . . . . . . . . . . . . . . . .. 73. 3.5.2.4. 10. 55. σ(I1 ) = I0 and σ(Iα ) = Iα , α = 2, 3, 4. 71.

(12) CONTENTS 4 Commutants in crossed products for algebras of piece-wise constant functions 79 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 4.2 Definitions and a preliminary result . . . . . . . . . . . . . . . . .. 80. 4.3 Algebra of piece-wise constant functions on the real line with N fixed jump points . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. 4.4 Comparison of commutants . . . . . . . . . . . . . . . . . . . . . .. 84. 4.4.1. An example . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86. 4.5 Description of the center . . . . . . . . . . . . . . . . . . . . . . . .. 88. 4.6 Jump points added into different intervals . . . . . . . . . . . . . .. 90. 4.6.1. Description of. SepnAS (R). and the commutant. AS. . . . . . .. 91. 5 Commutants in crossed product algebras for piece-wise constant functions on the real line 97 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 5.2 Definitions and a preliminary result . . . . . . . . . . . . . . . . .. 98. 5.2.1. Automorphisms induced by bijections . . . . . . . . . . . .. 99. 5.3 Commutants in crossed product algebras for piece-wise constant functions on the real line . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 Jump points added arbitrarily . . . . . . . . . . . . . . . . . . . . . 102 5.4.1. A condition for invariance. . . . . . . . . . . . . . . . . . . 102. 5.5 Finitely many jump points added . . . . . . . . . . . . . . . . . . . 103 5.5.1. Jumps added into different intervals . . . . . . . . . . . . . 103. 5.5.2. A comparison of the commutants . . . . . . . . . . . . . . . 104. 5.6 An example with two jump points added . . . . . . . . . . . . . . . 105 5.6.1. Jump points added into the same interval . . . . . . . . . . 106 5.6.1.1. σ(Iαj 0 ) = Iαj 0 for all j = 1, 2, · · · , 5 . . . . . . . . . 106. 5.6.1.2. σ(Iα10 ) = Iα20 , σ(Iα20 ) = Iα10 and σ(Iαj 0 ) = Iαj 0 , j = 3, 4, 5 . . . . . . . . . . . . . . . . . . . . . . . . . 106. 11.

(13) Dynamical Systems and Commutants in Non-Commutative Algebras. 5.6.2. 5.6.1.3. σ(Iα10 ) = Iα20 , σ(Iα20 ) = Iα30 , σ(Iα30 ) = Iα10 and σ(Iαj 0 ) = Iαj 0 for j = 4, 5 . . . . . . . . . . . . . . . 107. 5.6.1.4. σ(Iα10 ) = Iα20 , σ(Iα20 ) = Iα30 , σ(Iα30 ) = Iα10 , σ(Iα40 ) = Iα50 and σ(Iα50 ) = Iα40 . . . . . . . . . . . . . . . . . 107. Jump points added into different intervals . . . . . . . . . . 108 5.6.2.1. σ(si ) = si , i = 1, 2 and σ(Iαj i ) = Iαj i for all j = 1, 2, 3108. 5.6.2.2. σ(si ) = si for all i = 1, 2, σ(Iα11 ) = Iα21 , σ(Iα12 ) = Iα12 109. 5.6.2.3. σ(s1 ) = s2 (⇒ σ(s2 ) = s1 ) . . . . . . . . . . . . . . 109. 5.7 Comparison of commutants for general sets . . . . . . . . . . . . . 110 5.7.1. Partitioning one set . . . . . . . . . . . . . . . . . . . . . . 112. 5.7.2. Partitioning more than one set . . . . . . . . . . . . . . . . 112. 6 Ore extensions of function algebras. 117. 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Ore extensions. Basic preliminaries . . . . . . . . . . . . . . . . . . 118 6.3 Derivations on algebras of functions on a finite set . . . . . . . . . 119 6.4 Centralizers in Ore extensions for function algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4.1. Centralizer for the case Δ = 0 . . . . . . . . . . . . . . . . . 126. 6.4.2. Centralizer for the case Δ = 0 . . . . . . . . . . . . . . . . . 127. 6.4.3. Center of the Ore extension algebra . . . . . . . . . . . . . 129. 6.5 Infinite-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5.1. Existence of the non-unital Ore extension . . . . . . . . . . 131. 6.5.2. Centralizer for A . . . . . . . . . . . . . . . . . . . . . . . . 136. 6.5.3. 6.5.2.1. The case Δ = 0 . . . . . . . . . . . . . . . . . . . 136. 6.5.2.2. The case Δ = 0 . . . . . . . . . . . . . . . . . . . 137. Center of A[x, σ˜ , Δ] when Δ = 0 . . . . . . . . . . . . . . . 137. 6.6 The skew-Laurent ring A[x, x−1 ; σ˜ ] . . . . . . . . . . . . . . . . . . 138. 12.

(14) CONTENTS 6.7 The skew power series ring . . . . . . . . . . . . . . . . . . . . . . 139 6.7.1. Centralizer of A in the skew power series ring A[x; σ˜ ] . . . 140. 6.7.2. The center of the skew power series ring . . . . . . . . . . . 141. 13.

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(16) Chapter 1 Introduction This thesis treats two topics which are very closely related. The description of commuting elements (or commuting operators) in a representing operator algebra and the description of the maximal commutative subalgebra containing a given algebra A within the crossed product of A with the group of integers (Z) where the latter act via a composition automorphism. We give the details in the following sections and the corresponding chapters but before that let us recall a few definitions.. 1.1. Definitions and general notions. Definition 1.1.1. An algebra is a vector space A over a field F, which is equipped with a bilinear multiplication. That is, for all a, b, c ∈ A and λ, μ ∈ F (λa + μb)c = λac + μbc. and. a(λb + μc) = λab + μac.. An algebra A is called associative if the multiplication is associative, that is, a(bc) = (ab)c for every a, b, c ∈ A. Definition 1.1.2. Let A be an algebra. By two commuting elements, we mean elements a, b ∈ A such that ab = ba. An algebra A is commutative if every two elements a, b ∈ A commute. One item which will be studied extensively in this thesis is the commutant of a given subalgebra whose definition is given below.. 15.

(17) Dynamical Systems and Commutants in Non-Commutative Algebras Definition 1.1.3. Let B be any algebra and A ⊆ B be any subset (not necessarily a subalgebra) of B. The commutant of A, denoted by A , is defined as A = {b ∈ B : (∀ a ∈ A) ab = ba}. A simple remark is that the commutant (also known as the centralizer) of any subset of an algebra B is a subalgebra of B. In this study, we consider the algebra A of piece-wise constant functions, as a subalgebra of a crossed product. It turns out in this case that the commutant A is commutative and hence it is the maximal commutative subalgebra containing A. Definition 1.1.4. Let B be an algebra and let A be a commutative subalgebra of B. A is said to be maximal commutative if there is no other commutative subalgebra C of B such that A C. Equivalently, A is a maximal commutative subalgebra of B if and only if A is commutative and there exists no b ∈ B \ A such that ab = ba for all a ∈ A. Again this is equivalent to, A is maximal commutative if and only if A = A .. 1.2. Commuting elements in operator algebras associated to dynamical systems. This section is a background to Chapter 2 in which we give a description of commutativity of monomials of operators, satisfying certain commutation relations, in relation to periodic points of one-dimensional dynamical systems. The description of commuting elements in an algebra and commuting operators in a representing operator algebra, that is, the problem of explicit description of commutative subalgebras is central in representation theory and applications of non-commutative algebras. Commuting families of operators are a key ingredient in representation theory of many important algebras [2, 5, 9, 10, 11, 12, 18, 26, 27, 28, 29, 35, 36] and are important in the study of integrable systems and nonlinear equations [8, 19]. One famous example is that of linear operators A and B satisfying the commutation relation AB − BA = I,. (1.1). where I denotes the identity operator. Commutation relation (1.1) is known as the Heisenberg canonical commutation relation and elements A and B together generate the Weyl algebra or the Heisenberg algebra which is of interest in quantum mechanics among other areas. Relation (1.1) is satisfied, for example, by the rescaled by a constant operators of creation and annihilation for a system with one. 16.

(18) Commuting elements in operator algebras associated to dynamical systems degree of freedom. For a system with more than one degree of freedom, one of the most natural direct extensions of (1.1) consists of finite or infinite families of operators {Aj , Bj }j∈J satisfying the Heisenberg canonical commutation relations Aj B j − B j Aj = I Ai Bj − Bj Ai = 0 for i = j. (1.2). Ai Aj − Aj Ai = 0, Bi Bj − Bj Bi = 0 for all i, j ∈ J. If A = D : f(x) → (Df)(x) = f (x) is the operator of differentiation and B = Mx : f(x) → xf(x) is the operator of multiplication by the indeterminate x both acting say on the linear space of infinitely differentiable functions, or on the linear space of formal power series or polynomials in x, then they satisfy the Heisenberg canonical commutation relation (1.1). Similarly, the relations (1.2) are satisfied if for all j ∈ J = {1, 2, · · · , n} Aj = ∂j = ∂xj : f(x1 · · · xn ) → (∂xj f)(x1 , · · · , xn ) is the operator of partial differentiation with respect to the indeterminate xj , and Bj = Mxj : f(x1 , · · · , xn ) → xj f(x1 , · · · , xn ) is the operator of multiplication by xj , acting for example on the linear space of infinitely differentiable functions, or on the linear space of formal power series or polynomials in x1 , · · · , xn . These observations make the Heisenberg commutation relations fundamentally important for differentiation and integration theory. Therefore they play an important role in physics and many other subjects where integration and differentiation are involved. More about operators satisfying more general q−deformed Heisenberg commutation relations can be found in [8] and [13]. In this thesis, we consider linear operators on a Hilbert space satisfying more general commutation relations of the form. or. AB = F (BA). (1.3). XX ∗ = F (X ∗ X). (1.4). for which the Heisenberg commutation relation is a special case. In (1.4), X ∗ denotes the adjoint of X. Relations (1.3) and (1.4) are closely linked with covariance type relations of the form BA = F (A)B (1.5). 17.

(19) Dynamical Systems and Commutants in Non-Commutative Algebras or of the form AB = BF (A). (1.6). for some function F for which the expressions F (BA), F (X ∗ X) and F (A) make sense in terms of an appropriate functional calculus. In a purely algebraic context, the function F is usually a polynomial, X is an element of some associative ∗−algebra in case of (1.4) and A and B are elements of some associative algebra in the case of (1.3), (1.5) and (1.6). For operators on a Hilbert space or elements of a C ∗ −algebra or von Neumann algebra, F can be from broader classes of functions such as analytic, continuous or Borel measurable functions. We will call the pair of operators (X, X ∗ ) and (A, B) satisfying commutation relations (1.4), (1.5) and (1.6), a representation of the respective commutation relation. The most obvious way to see a close connection of (1.4) with (1.5) and (1.6) is that X(X ∗ X) = (XX ∗ )X = F (X ∗ X)X. Taking X = B and X ∗ X = A, yields (1.5). Also (X ∗ X)X ∗ = X ∗ (XX ∗ ) = X ∗ F (X ∗ X) which yields (1.6) if A = X ∗ X and B = X ∗ Commutation relations of the form (1.4),(1.5) and (1.6) play a central role in many directions in mathematics, quantum mechanics, statistical physics and quantum field theory [2, 5, 9, 10, 11, 13, 15, 18, 26, 27, 30, 31, 35]. The problem of determining the structure of commuting elements in algebras associated with commutation relations (1.4), (1.5) and (1.6) and connection of these commuting elements with periodic points of dynamical systems has been systematically studied in, for example, [19] and [31].. 1.2.1. An operator relation connected with dynamical systems. In this section, we give a construction of representations X in relation to dynamical systems and some results from [19] which form a basis for studies in Chapter 2. We start by recalling a few definitions. Let H be a complex Hilbert space. Definition 1.2.1. A linear operator X : H → H is said to be bounded if there exists a constant K 0 such that. X(x) K x. for all x ∈ H.. 18.

(20) Commuting elements in operator algebras associated to dynamical systems The set of all bounded linear operators on a Hilbert space H is denoted by B(H). Definition 1.2.2. The adjoint of a linear operator X ∈ B(H), denoted by X ∗ , is the (unique) linear operator X ∗ : H → H such that X(x), y

(21) = x, X ∗ (y)

(22) for all x, y ∈ H. Observe that X ∗ ∈ B(H) and X ∗ = X . Definition 1.2.3. An operator X ∈ B(H) is said to be 1. self-adjoint if X ∗ = X, 2. normal if X ∗ X = XX ∗ , 3. unitary if X ∗ X = XX ∗ = I where I is the identity operator on H. Definition 1.2.4. An orthogonal projection on H is a self-adjoint linear operator P : H → H such that P 2 = P. Definition 1.2.5. An operator X ∈ B(H) is called a partial isometry if. X(x) = x. for all x ∈ (ker X)⊥ , where for every subset W ⊆ H, W ⊥ = {x ∈ H : x, w

(23) = 0 for every w ∈ W }. This is equivalent to saying that X ∈ B(H) is a partial isometry if X ∗ is a generalized inverse of X, (XX ∗ X = X). Let X be a bounded linear operator on a Hilbert space H such that XX ∗ = F (X ∗ X). (1.7). where F : R → R is a function for which the expression (1.7) makes sense in terms of an appropriate functional calculus. Such an operator is called a representation or a bounded representation of the relation (1.7) and the pair of operators (X, X ∗ ) is called a ∗−representation of (1.7). Since X is bounded, there exists a polar decomposition X = U C such that C is a self-adjoint operator given by 1. C = C ∗ = (X ∗ X) 2. 19.

(24) Dynamical Systems and Commutants in Non-Commutative Algebras and U is a partial isometry. Furthermore, ker X = ker C = ker C 2 = ker U and U ∗ U is an orthogonal projection, denoted by P(ker C)⊥ on (ker C)⊥ . This decomposition and C = C ∗ imply that P(ker C)⊥ C = C. Hence, (U C)∗ U C = C ∗ U ∗ U C = C ∗ P(ker C)⊥ C = C ∗ C = C 2 and we can write (1.7) as, XX ∗ = F (X ∗ X) ⇔ U C(U C)∗ = F ((U C)∗ U C) ∗. (1.8). ⇔ U C U = F (C ) 2. 2. ⇔ U C = F (C )U, or 2. 2. (1.9) 2. ∗. ∗. 2. C U = U F (C ).. (1.10). The following Theorem gives the connection of the mapping F to the spectrum σ(C 2 ) of the operator C 2 = X ∗ X (see [19, 27, 28]). Theorem 1.2.1. Let C and U be such that (1.10) holds. If eλ is an eigenvector to C 2 with eigenvalue λ, then C 2 (U ∗ eλ ) = U ∗ (F (C 2 )eλ ) = F (λ)(U ∗ eλ ). Therefore, the vector U ∗ eλ is either zero or is an eigenvector to C 2 with eigenvalue F (λ). If X is invertible, that is, if U is unitary, then F (σ(C 2 )) ⊆ σ(C 2 ). Using Theorem 1.2.1 above construction of representations X of (1.7) can be done, in some specific cases, as follows. Take, if possible, a sequence of positive numbers {λk }k∈Z such that F (λk ) = λk+1 for all k ∈ Z. Such a sequence exists if, for example, we take λk to be a sequence of periodic points of F. Define for some orthonormal basis {ek }k∈N , the operators C and U in 2 (Z) so that U ek = ek−1 (1.11) C 2 e k = λk e k , for all k ∈ Z. Defined in this way, C and U satisfy (1.10) since U ∗ (ek ) = ek+1 and U C 2 ek = λk ek−1 F (C 2 )U ek = F (C 2 )ek−1 = F (λk−1 )ek−1 = λk ek−1 2 ∗ C U ek = λk+1 ek+1 U ∗ F (C 2 )ek = U ∗ F (λk )ek = λk+1 U ∗ ek = λk+1 ek+1 for all k ∈ Z. Thus by (1.8), for any C and U satisfying (1.11) the operator X = U C will be a representation of (1.7). It turns out that for some important class of mappings, any irreducible representation X of (1.7) with invertible X is unitarily equivalent to a representation X of this form [19, 27, 28].. 20.

(25) Commuting elements in operator algebras associated to dynamical systems. 1.2.2. Commutativity of monomials of operators on a finite-dimensional space and periodic orbits for one-dimensional dynamical systems. In this section we state commutativity conditions for monomials of operators A and B on a finite-dimensional space satisfying the commutation relation AB = BF (A) in terms of periodic points of F. First we give a few definitions. Definition 1.2.6. A point λ ∈ C is said to be a periodic point of F : C → C if F o(n) (λ) = λ for some integer n ≥ 1. The least such integer is called the period of the (periodic) point λ and in this case we say that λ is an n−periodic point. For an n−periodic point λ, we define the orbit of λ, Oλ , as Oλ = {F o(k) (λ) : k = 0, 1, · · · , n − 1}. Definition 1.2.7. For two operators A, B ∈ B(H), we define the commutator [A : B] of A and B as [A : B] = AB − BA. It follows immediately that A and B commute if and only if [A : B] = 0 Let F : C → C and ⎤ ⎡ ⎡ ⎤ 0 α λ 0 ⎥ ⎢1 0 ⎥ ⎢ ⎥ ⎢ (1.12) A = ⎣ ... , B = ⎥ ⎢ .. .. ⎦ ⎦ ⎣ . . o(n−1) 0 F (λ) 0 1 0 where |α| = 1 and λ ∈ P ern (F ), where P ern (F ) denotes the set of all n−periodic points of F. Then AB = BF (A), where F (A) is defined as applying F to each of the diagonal element in A. The following Theorem was proven in [19]. Theorem 1.2.2. Let A and B be of the form in (1.12) above. Let s ≡ s (mod n) and u ≡ u (mod n) be such that 0 u s < n. Then [B s At : B u Av ] = 0 if and only if F o(k) (λ)t F o(k+n−u ) (λ)v = F o(k+s −u ) (λ)v F o(k+n−u ) (λ)t for all k = 0, 1, · · · , n − 1, or equivalently F o(u) (μ)t μv = F o(s) (μ)v μt. (1.13). for all μ ∈ Oλ .. 21.

(26) Dynamical Systems and Commutants in Non-Commutative Algebras Motivated by the results in [19], the authors of that article suggested the following concrete problems concerned with dynamical systems which naturally arise in the context of Theorem 1.2.2 and have deep connections not only to the structure and representations of non commutative algebras and dynamics, but also to combinatorics, number theory and other areas of mathematics and mathematical physics. Consider the functional equations or a system of functional equations of the form (1.13), that is F o(u) (μ)t μv = F o(s) (μ)v μt where F : C → C and F o(n) , n ∈ Z are direct or inverse iterations. 1. The first problem is that of finding or studying the properties of the set of all μ ∈ C satisfying this equation for a given mapping F and choices of s, t, u, v ∈ Z. Especially, it is of interest to understand the intersection of this solution set with orbits of the system and in particular to describe the orbits which entirely belong to this set, if such orbits exist. 2. The second problem is to find or study the properties of those F in a given class of mappings, which satisfy (1.13) for all μ from a given subset of C. 3. Finally, the third problem is to find all integers s, t, u, v such that (1.13) is satisfied for a given F and for μ from a given subset of C. These questions form part of the motivation for our studies and the results in Chapter 2 provide solutions to problem 3 and some parts of problem 1 for some operators A and B on a finite-dimensional space and the case where (F, I) is the β−shift dynamical system on the interval I = [0, 1).. 1.3. Crossed product algebras and C ∗−crossed products. This section gives a background to chapters 3, 4 and 5 in which we study commutants in crossed products of algebras of piece-wise constant functions. Consider a pair (A, φ) where A is an arbitrary associative commutative complex algebra and φ : A → A is an automorphism. To this pair, we associate a crossed product containing an isomorphic copy of A, that is we define A φ Z := {f : Z → A : f(n) = 0 except for a finite number of n}.. (1.14). We endow it with the structure of an associative C−algebra by defining scalar multiplication and addition as the usual pointwise operations. Multiplication is. 22.

(27) Crossed product algebras and C ∗ −crossed products defined by convolution twisted by the automorphism φ as follows;. f(k)φk (g(n − k)), (fg)(n) =. (1.15). k∈Z. f, g ∈ A φ Z, n ∈ Z and φk denotes the k−fold composition of φ with itself for. k positive k ∈ Z and we use φ−k = φ−1 for k 0. Definition 1.3.1. A ×φ Z is called the crossed product of A and Z under φ and A is called the coefficient algebra. A useful and convenient way of

(28) working with A φ

(29) Z, is to write elements f, g ∈ A φ Z in the form f = n∈Z fn δ n and g = n∈Z gm δ m where fn = f(n), gm = g(m) and δ n = χ{n} , where for n, k ∈ Z,. 1, if k = n χ{n} (k) := (1.16) 0, if k = n. δ as defined in (1.16) satisfies the relation δf = φ(f)δ, for every f ∈ A. Addition and scalar multiplication are canonically defined by the usual pointwise operations and multiplication is determined by the relation (fn δ n ) ∗ (gm δ m ) = fn φn (gm )δ n+m ,. (1.17). where m, n ∈ Z and fn , gm ∈ A. Many studies have been carried out on A φ Z for various coefficient algebras about maximal commutative subalgebras, ideal intersections and many other properties in relation to the dynamics of φ [5, 18, 31, 32, 34, 36, 37]. One example that has been studied in detail is the C ∗ −crossed product algebra. Below we recall the definition of a C ∗ −algebra and give an important result connecting maximal commutativity of C(X) as a subalgebra of a C ∗ − crossed product and density of aperiodic points of X. Definition 1.3.2. An algebra A equipped with a norm is called a normed algebra if the norm is submultiplicative; that is ||ab|| a. b for all a, b ∈ A.. (1.18). If A is unital (that is, A has a ring identity 1A ), then we assume that ||1A || = 1.. 23.

(30) Dynamical Systems and Commutants in Non-Commutative Algebras Definition 1.3.3. A normed algebra which is complete in the metric induced by the norm is called a Banach algebra. Definition 1.3.4. A Banach ∗−algebra is a complex Banach algebra A endowed with a conjugate linear involution ∗ (also called the adjoint) which is an antiisomorphism, that is for all a, b ∈ A and λ ∈ C, BA1: (a + b)∗ = a∗ + b∗ ¯ ∗ BA2: (λa)∗ = λa BA3: a∗∗ = a BA4: (ab)∗ = b∗ a∗ . Associative algebras with involution are called involutive algebras or ∗ − algebras. In a Banach ∗−algebra (or in any normed involutive algebra) A, ||a∗ || = ||a|| for every a ∈ A.. (1.19). From (1.18) and (1.19) we have that ||a∗ a|| ||a||2 for every a ∈ A. Definition 1.3.5. A C∗ −algebra is a Banach ∗−algebra A with the additional norm condition (1.20) ||a∗ a|| = ||a||2 for all a ∈ A called the C ∗ property. Some examples of C ∗ − algebras include • B(H), the algebra of linear operators on a Hilbert space H with pointwise addition and scalar multiplication. Multiplication is defined by composition, the norm is given by ||X|| := sup {||X(x)|| : ||x|| 1} and the involution is defined as the operation of taking adjoint of an operator X → X ∗ ∈ B(H). • C(X) the algebra of continuous complex-valued functions on a compact topological space X with the usual pointwise operations (addition, scalar multiplication and multiplication) and norm given by ||f|| := sup {|f(x)| : x ∈ X}.. 24.

(31) Crossed product algebras and C ∗ −crossed products As mentioned before, the class of C ∗ −algebras that has been extensively studied is the class of C ∗ − crossed product algebras. One starts with a topological dynamical system Σ = (X, σ), where X is a compact Hausdorff topological space and σ is a homeomorphism of X and integers act on X via iterations of σ. A = C(X) is the algebra of continuous complex-valued functions on X endowed with the supremum norm and the natural pointwise operations. The automorphism φ : A → A is defined for all f ∈ A by φ(f) = f ◦ σ −1 . C(X) φ Z is endowed with a structure of an algebra as defined in (1.14). The involution ∗ is defined by f ∗ (n) = φn (f(−n)) for f ∈ A φ Z and n ∈ Z, where the bar denotes the usual pointwise complex conjugation. A = C(X) is a ∗−subalgebra of A φ Z, namely as {f : Z → A : (∀ n = 0) f(n) = 0}. The norm on A φ Z is defined as follows. For f ∈ A φ Z,. fn δ n , f= n. where the fn ∈ A and δ is unitary, that is, δ ∗ = δ −1 ,. fn ∞ .. f = n. Completing A φ Z in this norm yields a Banach ∗−algebra 1 (Σ). That is. . 1 n fn δ : fn ∈ A and. fn ∞ < ∞ , (Σ) = f = n. n. with the operations of A φ Z extended by continuity. The C ∗ −crossed product, C ∗ (Σ), associated to the dynamical system Σ is the enveloping algebra of 1 (Σ) in another norm. Details of this can be found in [33, 36]. This C ∗ −algebra is usually also denoted by C(X) φ Z. For a dynamical system Σ = (X, σ), a point x ∈ X is called aperiodic if σ n (x) = x for every nonzero n ∈ Z. The system Σ is called topologically free if the set of aperiodic points is dense in X. The following Theorem which appears as Theorem 4.3.5 in [36] states the equivalence between density of aperiodic points of X and maximal commutativity of C(X).. 25.

(32) Dynamical Systems and Commutants in Non-Commutative Algebras Theorem 1.3.1. The following properties are equivalent. 1. Σ is topologically free. 2. Every non-zero closed ideal I of C(X) φ Z is such that I ∩ C(X) = {0}. 3. C(X) is a maximal abelian C ∗ −subalgebra of C(X) φ Z. Several investigations have between carried out on the interplay between the C ∗ −dynamical system Σ and the C ∗ −crossed product C(X)φ Z and more general C ∗ −crossed product algebras [3, 5, 18, 26, 31, 32, 34, 36, 37]. In [34], a slightly different approach was taken. Connections between properties of commutative subalgebras of crossed product algebras and properties of dynamical systems that are in many situations naturally associated with the construction were brought into an algebraic context. This consists of the construction of a crossed product A φ Z of an arbitrary subalgebra A of the algebra CX of functions on a set X (under the usual pointwise operations) by Z, where the latter acts on A by a composition automorphism. This algebraic framework allows one to investigate the relation between the maximality of the commutative subalgebra in the crossed product on one hand and the properties of the action on the space on the other hand, for arbitrary choices of the set X, the subalgebra A and the action, different from the previously cited classical choice of continuous functions C(X) on a compact Hausdorff topological space X. However, in this algebraic framework, topological notions are lost so the condition of topological freeness of the dynamical system as described above is not applicable anymore. Therefore it has to be generalized in a proper way in order to be equivalent to maximal commutativity of A. This forms the motivation of our studies in this thesis where we consider crossed product algebras for the algebra AX of functions that are constant on partitions of a set X. The algebra AX is not necessarily invariant under any bijection σ : X → X so appropriate conditions have to be derived such that (X, σ) is a dynamical system.. 1.3.1. Automorphisms induced by bijections. Let X be a non-empty set, σ : X → X a bijection on X and A ⊆ CX be an algebra of functions that is invariant under σ and σ −1 , that is, if h ∈ A, then h ◦ σ and h ◦ σ −1 ∈ A. Then (X, σ) is a discrete dynamical system. The action of n ∈ Z on x ∈ X is given by n : x → σ n (x) and σ induces an automorphism σ˜ : A → A defined by σ˜ (f) = f ◦ σ −1 by which Z acts on A via iterations. In this thesis, we shall treat the crossed algebra A σ˜ Z for this setting, where A is the algebra of functions that are constant on the partitions of X. Below are generalizations of periodic and aperiodic points of σ as described in [34].. 26.

(33) Crossed product algebras and C ∗ −crossed products Definition 1.3.6. For any nonzero n ∈ Z, we set SepnA (X) = {x ∈ X : (∃ h ∈ A such that) h(x) = σ˜ n (h)(x)}, n P erA (X) = {x ∈ X : (∀ h ∈ A) h(x) = σ˜ n (h)(x)},. Sepn (X) = {x ∈ X : x = σ n (x)}, P ern (X) = {x ∈ X : x = σ n (x)}. Furthermore, let ∞ (X) = P erA. . SepnA (X).. n∈Z\0. P er∞ (X) =. . Sepn (X).. n∈Z\0. Finally, for f ∈ A, set supp(f) = {x ∈ X | f(x) = 0}. Then all these sets, except supp(f), are Z−invariant and if A separates points of X, for example, if A = C(X), the algebra of continuous complex-valued functions on a compact Hausdorff topological space X, then SepnA (X) = Sepn (X) and n (X) = P er n (X). Note also that Sepn (X) = X \ P er n (X) and Sepn (X) = P erA A A X \ P ern (X). Furthermore SepnA (X) = Sep−n A (X) with similar equalities for n n (X), Sepn (X) and P er n (X) as well. It should and −n, (n ∈ Z), holding for P erA be noted also that the algebras under consideration in this thesis do not separate points so we are working in this algebraic framework. The following two important results appear in [34] as Theorem 2.3.3 and Corollary 2.3.4 respectively. Theorem 1.3.2. The unique maximal abelian subalgebra of A σ˜ Z that contains A is precisely the set of elements. . A =. fn δ. n. : (∀ n ∈ Z) fn |SepnA (X) ≡ 0 .. n∈Z. Corollary 1.3.1. If A separates points of X, then A is precisely the set of elements. . fn δ n : (∀ n ∈ Z) supp(fn ) ⊆ P ern (X) . A = n∈Z. 27.

(34) Dynamical Systems and Commutants in Non-Commutative Algebras. 1.4. Ore extensions. In Chapter 6, we shift our attention to the treatment of commuting elements in Ore extensions of functional algebras. Ore extensions were first studied by a Norwegian mathematician Øystein Ore [17]. An Ore extension of a ring A is an overring with a generator x satisfying xa = σ(a)x + Δ(a) for a ∈ A, σ an endomorphism of A and Δ a σ−derivation on A. For an overall description of Ore extensions one can see, for example, [6, 14, 22]. Here we give a few definitions. Definition 1.4.1. Let A be an associative ring and σ : A → A a ring endomorphism (not necessarily injective). A σ-derivation is a map Δ : R → R such that Δ(a + b) = Δ(a) + Δ(b) and Δ(ab) = σ(a)Δ(b) + Δ(a)b for all a, b ∈ A. Note that if Δ is a σ−derivation and A is unital, (we assume all morphisms are unital) then Δ(1) = Δ(1 · 1) = σ(1)Δ(1) + Δ(1)1 = 2Δ(1), from which we get that Δ(1) = 0. Definition 1.4.2. For any endomorphism σ : A → A and any a ∈ A, define Δ : A → A by Δ(b) = ab − σ(b)a. Then Δ is a σ−derivation. All derivations of this form are called inner σ−derivations. Those σ−derivations which are not inner are called outer. Definition 1.4.3. Let A be a ring, σ an endomorphism of A and Δ a σ−derivation. The Ore extension A[x; σ, Δ] is the ring of polynomials over A equipped with a new multiplication, satisfying xa = σ(a)x + Δ(a). (1.21). for all a ∈ A. By (1.21) multiplication is defined uniquely. It has been proven [16, 22, 24] that for any associative ring A, the Ore extension A[x; σ, Δ] is well-defined if and only if σ is an endomorphism of R and Δ is a σ−derivation of R. Definition 1.4.4. An element a ∈ A is a scalar of the Ore extension A[x, σ, Δ] if σ(a) = a and Δ(a) = 0.. 28.

(35) Ore extensions The set of scalars forms a subring of any Ore extension. If σ = idR , then we say that A[x; idR , Δ] is a differential polynomial ring. If instead Δ ≡ 0, then we say that A[x; σ, 0] is a skew polynomial ring. It is important to note that some authors use the term skew polynomial rings to mean Ore extensions.. 1.4.1. Centralizers in Ore extensions. An Ore extension A[x; σ, Δ] is commutative if and only if A is commutative, σ = id and Δ = 0. Therefore, it is an interesting question to investigate the centralizers of elements in Ore extensions. Definition 1.4.5. If A is a ring and S ⊆ A, then the centralizer of S, denoted by C(S) is defined as C(S) := {a ∈ A : (∀ s ∈ S) as = sa}. Several studies have been done in this direction [1, 4, 7, 23, 38]. In [38], a description of the centralizer of a commutative ring A in the Ore extension A[x, σ, Δ] and conditions for when A is maximal commutative are given. Note that if S is a commutative subring of A, then S is maximal commutative if the centralizer C(S) of S in A coincides with S. The following Theorem, which gives the description of the centralizer of A in the Ore extension A[x, σ, Δ], appears as Proposition 3.1 in [38]. Theorem 1.4.1. An element if and only if. m

(36). fk xk ∈ A[x, σ˜ , Δ] belongs to the centralizer of A. k=0. gfk =. m. fj πkj (g). (1.22). j=k. holds for all k ∈ {0, 1, · · · , m} and all g ∈ A, where the functions πkl : A → A, for k, l ∈ Z are defined as follows; π00 = id. If m, n are nonzero integers such n = 0. For the other that m > n or atleast one of the integers is negative, then πm remaining cases, n−1 n n−1 πm = σ˜ ◦ πm−1 + Δ ◦ πm . Motivated by our results in Chapters 3 − 5 and using this Theorem, we give a description of the centralizer of A in the Ore extension A[x, σ, Δ], in the case when A is the algebra of functions on a countable set with finite support in Chapter 6.. 29.

(37) Dynamical Systems and Commutants in Non-Commutative Algebras. 1.5. Summary of the thesis. Chapter 2 up to Chapter 6 correspond, respectively, to papers A, B, C, D and E, whose contents we summarize below.. 1.5.1. On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems. In Chapter 2, we give an explicit description of how periodic points of F affect commutativity of the monomials Φ = B s At and G = B u Av for the linear operators A and B on a finite-dimensional Hilbert space defined as follows Aej = λj ej j = 0, 1, · · · , n − 1, Bej = ej+1 j = 0, 1, · · · , n − 2, Ben−1 = e0 . It should be noted that these operators are of the form (1.11) and therefore they satistfy the relation AB = BF (A) where F (A) is defined as applying F onto each of the diagonal elements of A. We consider the case where {λj }nj=1 are periodic points of F, then F o(n) (A) = A. Explicit conditions on the integers s, t, u and v for commutativity of the monomials Φ = B s At and G = B u Av are stated in terms of existence of periodic points of F with periods that are integer divisors of n. Finally, we apply the results to the special case when (F, I) is the β−shift dynamical system on the interval I = [0, 1), where we also derive conditions on β for existence of periodic points of period n for any n ∈ Z.. 1.5.2. Crossed product algebras for piece-wise constant functions. Let X beany set, J a countable set and P = {Xj : j ∈ J} a partition of X; that Xr where Xr = ∅ for all r ∈ J and Xr ∩ Xr = ∅ if r = r . is X = r∈J. Let A be the algebra of piece-wise constant, complex-valued functions on X. That is A = h ∈ CX : (∀ j ∈ J) h(Xj ) = {cj } . In Chapter 3, we consider the crossed product algebra A σ˜ Z, where σ˜ is the automorphism of A induced by a bijection σ : X → X. We first derive conditions under which A is invariant under any bijection σ : X → X and then describe. 30.

(38) Summary of the thesis the crossed product algebra 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.. 1.5.3. Commutants in crossed products for algebras of piece-wise constant functions. In Chapter 4, we consider commutants in crossed product algebras for algebras of piece-wise constant functions on the real line with Z. Starting with an algebra At1 ,··· ,tN of piece-wise constant functions with N fixed jumps at points t1 , · · · , tN (with ti < tj if i < j and N 1), we extend it to an algebra At1 ,··· ,tN+M of piecewise constant functions with at most N + M jumps. It follows then that At1 ,··· ,tN is a subalgebra of At1 ,··· ,tN+M and therefore At1 ,··· ,tN+M is a subalgebra of At1 ,··· ,tN for every M 1. We describe the difference in the commutants At1 ,··· ,tN \ At1 ,··· ,tN+M . We also give a description of the center of the crossed product algebra A σ˜ Z where A is an algebra of piece-wise constant functions with N fixed jump points on the real line. This center happens to be the commutant of a subset of the crossed product algebra At1 ,··· ,tN+M σ˜ Z which happens to be a generator of the whole crossed product.. 1.5.4. Commutants in crossed product algebras for piecewise constant functions on the real line. Chapter 5 treats in a more general case, what was studied in Chapter 4. Starting with an algebra A of piece-wise constant functions with N fixed jump points, we add a finite number of jumps, say m arbitrarily and consider the algebra AS of piece-wise constant functions with N + m jumps. We derive a condition for the algebras A and AS to be invariant under a bijection σ : R → R and compare the commutants A and AS .. 1.5.5. Ore extensions of function algebras. Chapter 6 aims at giving a description of the centralizer and the center of the coefficient subalgebra A in the Ore extension algebra A[x, σ˜ , Δ], where A is the algebra of functions with finite support on a countable set X and σ˜ is an automorphism of A that is induced by a bijection σ : X → X. We give a description of twisted. 31.

(39) Dynamical Systems and Commutants in Non-Commutative Algebras derivations on the algebra of functions on a finite set from which it is observed that there are no non-trivial derivations on Rn , and describe the centralizer of the coefficient algebra A and the center of the Ore extension A[x, σ˜ , 0]. We also treat the case when A is the algebra of functions with finite support on a countable set, give a description for the centralizer and the center of the Ore extension A[x, σ˜ , 0].. References [1] Amitsur, S. A., (1958). Commutative linear differential operators, Pacific J. Math. 8, 1–10. [2] Bratteli, O., Robinson, D., (1981). Operator algebras and statistical mechanics, Springer-Verlag, ISBN: 0-387-10381-3 [3] Carlsen, T. M., Silvestrov, S. D., (2009). On the Exel crossed product of topological covering maps, Acta Appl. Math. 108, no. 3, 573-583. [4] Carlson, R., C., Goodearl, K., R., (1980). Commutants of ordinary differential operators, J. Differential equations. 35, no. 3, 339–365. [5] Davidson, K. R., (1996). C*-Algebras by example, American Mathematical Society. [6] Goodearl, K., R., Warfield, R., B., (2004). An introduction to non commutative Noetherian rings. Second edition. London Mathematical Society Student Texts, 61. Cambridge University Press, Cambridge. [7] Goodearl, K. R., (1983). Centralizers in differential, pseudo differential and fractional differential operator rings, Rocky Mountain J. Math. 13, no.4, 573–618. [8] Hellström, L., Silvestrov, S. D., (2000). Commuting elements in q−deformed Heiseberg algebras, World Scientific, pp. 256. ISBN:981-024403-7 [9] Li, B. R., (1992). Introduction to operator algebras, World Scientific. [10] Mackey, G.W., (1989).Unitary group representations in physics, probability and number theory, Addison-Wesley. [11] Mackey, G. W., (1976). The theory of unitary group representations, The University of Chicago Press.. 32.

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