On one-dimensional dynamical systems and commuting elements in non-commutative algebras

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Mälardalen University Press Licentiate Theses No. 234

ON ONE-DIMENSIONAL DYNAMICAL SYSTEMS AND

COMMUTING ELEMENTS IN NON-COMMUTATIVE ALGEBRAS

Alex Behakanira Tumwesigye

2016

School of Education, Culture and Communication

Mälardalen University Press Licentiate Theses

No. 234

ON ONE-DIMENSIONAL DYNAMICAL SYSTEMS AND

COMMUTING ELEMENTS IN NON-COMMUTATIVE ALGEBRAS

Alex Behakanira Tumwesigye

2016

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ISSN 1651-9256

Printed by Arkitektkopia, Västerås, Sweden

This thesis work is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. In Mathemat-ics, it is well known that matrix multiplication (or composition of linear operators on a finite dimensional vector space) is not always commutative. Commutating matrices or more general linear or non-linear operators play an essential role in Mathematics and its applications in Physics, and Engineer-ing. Many important relations in Mathematics, Physics and Engineering are represented by operators satisfying a number of commutation relations. Such commutation relations play 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 opera-tors 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 de-scribe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this al-gebra. In Chapter 4, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing com-mutants for a non decreasing sequence of algebras.

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This thesis work is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. In Mathemat-ics, it is well known that matrix multiplication (or composition of linear operators on a finite dimensional vector space) is not always commutative. Commutating matrices or more general linear or non-linear operators play an essential role in Mathematics and its applications in Physics, and Engineer-ing. Many important relations in Mathematics, Physics and Engineering are represented by operators satisfying a number of commutation relations. Such commutation relations play 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 opera-tors 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 de-scribe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this al-gebra. In Chapter 4, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing com-mutants for a non decreasing sequence of algebras.

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Acknowledgements

First and foremost, I would like to thank my supervisor Sergei Silvestrov who

accepted to take me up as his first student under 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

Johan Richter for his valuable and constructive suggestions during the

numer-ous academic discussions we had. I learnt 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 Linus Carlsson for the academic engagements we had.

I thank my supervisor from Uganda, 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 daddy has been away

in Sweden pursuing PhD studies. It has been comforting to know that I

could count on your support through out 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 on, 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 International

Science Programme (ISP) and the East African Universities Mathematics

Programme (EAUMP) for the financial support that facilitated my travels

and stay in Sweden. In a special way, I thank Pravina and Leif at ISP,

Uppsala for always providing quick answers and ensuring a comfortable stay

in Sweden.

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Acknowledgements

First and foremost, I would like to thank my supervisor Sergei Silvestrov who

accepted to take me up as his first student under 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

Johan Richter for his valuable and constructive suggestions during the

numer-ous academic discussions we had. I learnt 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 Linus Carlsson for the academic engagements we had.

I thank my supervisor from Uganda, 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 daddy has been away

in Sweden pursuing PhD studies. It has been comforting to know that I

could count on your support through out 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 on, 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 International

Science Programme (ISP) and the East African Universities Mathematics

Programme (EAUMP) for the financial support that facilitated my travels

and stay in Sweden. In a special way, I thank Pravina and Leif at ISP,

Uppsala for always providing quick answers and ensuring a comfortable stay

in Sweden.

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and Operator Algebras

I would like to thank the staff at the school of education, culture and

communication, M¨alardalens 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 from IDT, Milica Rancic, Maria Larsson and Marie Bergman from

UKK, fellow PhD students under ISP, Betuel Jesus Canhanga, Carole Ogutu,

Jean-Paul Murara and all other PhD students in Mathematics and Applied

Mathematics, M¨alardalens university for the nice moments we shared.

Finally, I thank my colleagues in the Department of Mathematics,

Mak-erere University for being such a wonderful family.

V¨aster˚

as, May, 2016

Alex Behakanira Tumwesigye

List of Papers

The chapters 2, 3 and 4 in this thesis are based, respectively, on the following papers:

Paper A. Tumwesigye, A. B., Silvestrov, S. D. (2014), On monomial commutativity

of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems, AIP Conference Proceedings, 1637,

1110-1119.

Paper B. Tumwesigye, A. B., Richter, J., Silvestrov, S. D., Ssembatya V. A. (2016).

Crossed product algebras for piecewise constant functions, to appear in

Silve-strov S., Ranˇci´c M., (eds.), Engineering Mathematics and algebraic, analysis and stochastic structures for networks, data classification and optimization, Springer Proceedings in Mathematics and Statistics, Springer, pp. 18. Paper C. Tumwesigye, A. B., Richter, J., Silvestrov, S. D., (2016). Commutants in

crossed products for algebras of piecewise constant functions, to appear in

Silvestrov S., Ranˇci´c M., (eds.), Engineering Mathematics and algebraic, analysis and stochastic structures for networks, data classification and opti-mization, Springer Proceedings in Mathematics and Statistics, Springer, pp. 12.

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and Operator Algebras

I would like to thank the staff at the school of education, culture and

communication, M¨alardalens 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 from IDT, Milica Rancic, Maria Larsson and Marie Bergman from

UKK, fellow PhD students under ISP, Betuel Jesus Canhanga, Carole Ogutu,

Jean-Paul Murara and all other PhD students in Mathematics and Applied

Mathematics, M¨alardalens university for the nice moments we shared.

Finally, I thank my colleagues in the Department of Mathematics,

Mak-erere University for being such a wonderful family.

V¨aster˚

as, May, 2016

Alex Behakanira Tumwesigye

List of Papers

The chapters 2, 3 and 4 in this thesis are based, respectively, on the following papers:

Paper A. Tumwesigye, A. B., Silvestrov, S. D. (2014), On monomial commutativity

of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems, AIP Conference Proceedings, 1637,

1110-1119.

Paper B. Tumwesigye, A. B., Richter, J., Silvestrov, S. D., Ssembatya V. A. (2016).

Crossed product algebras for piecewise constant functions, to appear in

Silve-strov S., Ranˇci´c M., (eds.), Engineering Mathematics and algebraic, analysis and stochastic structures for networks, data classification and optimization, Springer Proceedings in Mathematics and Statistics, Springer, pp. 18. Paper C. Tumwesigye, A. B., Richter, J., Silvestrov, S. D., (2016). Commutants in

crossed products for algebras of piecewise constant functions, to appear in

Silvestrov S., Ranˇci´c M., (eds.), Engineering Mathematics and algebraic, analysis and stochastic structures for networks, data classification and opti-mization, Springer Proceedings in Mathematics and Statistics, Springer, pp. 12.

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Chapter 1 Introduction

This thesis treats two very closely related topics; The description of commuting elements (or commuting operators) in a representing operator algebra and the description of the maximal commutative subalgebra containing a given algebraA

within the crossed product ofA 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 Some definitions and general notions

Definition 1.1.1. LetA 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.

Definition 1.1.2. LetB be any algebra and A ⊆ B be any subset (not necessarily

a subalgebra) ofB. The commutant of A, denoted by A, is defined as A={b ∈ B | ab = ba for all a ∈ A}

A simple remark is that the commutant of any subset of an algebra B is a

subalgebra ofB. In this study, we consider the algebra A of piecewise constant

functions, as a subalgebra of a crossed product. It turns out that the commutant

A_{is commutative and hence, it is the maximal commutative subalgebra containing}

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Chapter 1 Introduction

This thesis treats two very closely related topics; The description of commuting elements (or commuting operators) in a representing operator algebra and the description of the maximal commutative subalgebra containing a given algebraA

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 Some definitions and general notions

Definition 1.1.1. Let A be an algebra. By two commuting elements, we mean

elements a, b∈ A such that ab = ba.

An algebraA 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.

Definition 1.1.2. LetB be any algebra and A ⊆ B be any subset (not necessarily

a subalgebra) ofB. The commutant of A, denoted by A, is defined as A={b ∈ B | ab = ba for all a ∈ A}

A simple remark is that the commutant of any subset of an algebra B is a

subalgebra of B. In this study, we consider the algebra A of piecewise constant

functions, as a subalgebra of a crossed product. It turns out that the commutant

A_{is commutative and hence, it is the maximal commutative subalgebra containing}

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and Operator Algebras

Definition 1.1.3. 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

subalgebraC of B such that A C.

Equivalently, A is a maximal commutative subalgebra of B if there exists no b∈ B \ A such that ab = ba for all a ∈ A. Again this is equivalent to, A is maximal

commutative ifA = A, whereAdenotes the commutant ofA.

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 commu-tativity of monomials of operators in relation to periodic points on one-dimensional dynamical systems.

The description of commuting elements in an algebra and commuting opera-tors 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 [12, 16, 17, 21] and are im-portant in the study of integrable systems and non-linear equations [21, 24]. 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. Relations (1.1) are satisfied for example by the operators of creation and annihilation for a system with one degree of freedom. For a system with more than one degree of freedom, one considers instead finite or infinite families of operators {Aj, Bj}j∈J satisfying the Heisenberg canonical commutation relations

AjBj− BjAj = I

AiBj− BjAi = 0 for i= j (1.2)

AiAj− AjAi = 0, BiBj− BjBi= 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

dynamical systems

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)→ (∂xjf )(x1,· · · , xn)

is the operator of partial differentiation with respect to the indeterminate xj, and

Bj= Mxj: f (x1,· · · , xn)→ xjf (x1,· · · , xn)

is the operator of multiplication by xj, acting for example on the linear space of infinitely differentiable functions, or on the linear spaces of formal power series or polynomials in x1,· · · , xn. These observations make the Heisenberg commuta-tion relacommuta-tions fundamentally important for differentiacommuta-tion and integracommuta-tion theory. Therefore they play an important role in physics and many other subjects where integration and differentiation are involved. More about operators satisfying a more general q−deformed Heisenberg commutation relation can be found in [24].

In this thesis, we consider linear operators on a Hilbert space satisfying more general commutation relations of the form

AB = F (BA) (1.3) or

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) 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 con-text, 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), respectively, a representation of the respective commutation relation.

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and Operator Algebras

Definition 1.1.3. 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

subalgebraC of B such that A C.

Equivalently, A is a maximal commutative subalgebra of B if there exists no b∈ B \ A such that ab = ba for all a ∈ A. Again this is equivalent to, A is maximal

commutative ifA = A, whereA denotes the commutant ofA.

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 commu-tativity of monomials of operators in relation to periodic points on one-dimensional dynamical systems.

The description of commuting elements in an algebra and commuting opera-tors 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 [12, 16, 17, 21] and are im-portant in the study of integrable systems and non-linear equations [21, 24]. 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. Relations (1.1) are satisfied for example by the operators of creation and annihilation for a system with one degree of freedom. For a system with more than one degree of freedom, one considers instead finite or infinite families of operators {Aj, Bj}j∈J satisfying the Heisenberg canonical commutation relations

AjBj− BjAj = I

AiBj− BjAi = 0 for i= j (1.2)

AiAj− AjAi = 0, BiBj− BjBi= 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

dynamical systems

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)→ (∂xjf )(x1,· · · , xn)

is the operator of partial differentiation with respect to the indeterminate xj, and

Bj= Mxj: f (x1,· · · , xn)→ xjf (x1,· · · , xn)

is the operator of multiplication by xj, acting for example on the linear space of infinitely differentiable functions, or on the linear spaces of formal power series or polynomials in x1,· · · , xn. These observations make the Heisenberg commuta-tion relacommuta-tions fundamentally important for differentiacommuta-tion and integracommuta-tion theory. Therefore they play an important role in physics and many other subjects where integration and differentiation are involved. More about operators satisfying a more general q−deformed Heisenberg commutation relation can be found in [24].

In this thesis, we consider linear operators on a Hilbert space satisfying more general commutation relations of the form

AB = F (BA) (1.3) or

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) 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 con-text, 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), respectively, a representation of the respective commutation relation.

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and Operator Algebras

The most obvious way to see the close connection of (1.4) with (1.5) and (1.6) is that

X(X∗X) = (XX∗)X = F (X∗X)X,

therefore 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 [4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 27].

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 was first systematically studied in [19] and [21]. Below we give a construction of represen-tations of X in relation to dynamical systems and some results from [21] which form a basis for our studies in Chapter 2.

1.2.1 An operator relation connected with dynamical

systems

Let X be a bounded linear operator on a Hilbert spaceH such that

XX∗= F (X∗X) (1.7)

where F is a measurable function for which the expression (1.7) makes sense in terms of an appropriate functional calculus. Such an operator is called a represen-tation or a bounded represenrepresen-tation 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 C = C∗_{= (X}∗_X)12, and U is a partial isometry. Furthermore, ker X = ker C = ker C2= ker U and U∗U is an orthogonal projection 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 = C2 and we can write

(1.7) as,

XX∗= F (X∗X)⇔ UC(UC)∗= F ((U C)∗U C) (1.8)

⇔ UC2_U∗_{= F (C}2₎ _(1.9)

⇔ UC2_{= F (C}2_{)U, or} _C2_U∗_{= U}∗_{F (C}2_). _(1.10)

The following Theorem which appears as Theorem 2.2 in [21] gives the connection of the mapping F to the spectrum σ(C2_{) of the operator C}2_{= X}∗_X.

dynamical systems

Theorem 1.2.1. Let C and U be such that (1.10) holds. If eλ is an eigenvector

to C2with eigenvalue λ, then C2(U∗eλ) = U∗(F (C2)eλ) = F (λ)(U∗eλ). Thus, the

vector U∗eλ is either zero or is an eigenvector to C2 with eigenvalue F (λ).

If X is invertible, that is, if U is unitary, then F (σ(C2₎₎_{⊆ σ(C}2_).

Using Theorem 1.2.1 above construction of representations X of (1.7) can be done as follows.

Take, if possible, a sequence of positive numbers{λk}k∈Zsuch that

F (λk) = λk+1

for all k∈ Z. Define for some orthonormal basis {ek}k∈N, the operators C and U in 2(Z) so that

C2ek= λkek, U ek= ek−1 (1.11) for all k∈ Z. Defined in this way, C and U satisfy (1.10) since U∗_(e_k_{) = e}_k+1 _and

U C2_e k= λkek−1 F (C2)U ek= F (C2)ek−1= F (λk−1)ek−1= λkek−1 C2_U∗_e_k_{= λ}_k+1_e_k+1 U∗F (C2)ek= U∗F (λk)ek= λk+1U∗ek= λk+1ek+1

for all k∈ Z. Thus by (1.8), for any C and U satisfying (1.11) the operator X = UC

will be a representation of (1.7). It turns out that for some important important class of mappings, any irreducible representation X of (1.7) with invertible X is unitarily equivalent to a representation X of this form [21].

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 1.2.1. A point λ∈ C is said to be a periodic point of F if Fo(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

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and Operator Algebras

The most obvious way to see the close connection of (1.4) with (1.5) and (1.6) is that

X(X∗X) = (XX∗)X = F (X∗X)X,

therefore 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 [4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 27].

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 was first systematically studied in [19] and [21]. Below we give a construction of represen-tations of X in relation to dynamical systems and some results from [21] which form a basis for our studies in Chapter 2.

1.2.1 An operator relation connected with dynamical

systems

Let X be a bounded linear operator on a Hilbert space H such that

XX∗= F (X∗X) (1.7)

where F is a measurable function for which the expression (1.7) makes sense in terms of an appropriate functional calculus. Such an operator is called a represen-tation or a bounded represenrepresen-tation 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 C = C∗_{= (X}∗_X)12, and U is a partial isometry. Furthermore, ker X = ker C = ker C2= ker U and U∗U is an orthogonal projection 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 = C2 and we can write

(1.7) as,

XX∗= F (X∗X)⇔ UC(UC)∗= F ((U C)∗U C) (1.8)

⇔ UC2_U∗_{= F (C}2₎ _(1.9)

⇔ UC2_{= F (C}2_{)U, or} _C2_U∗_{= U}∗_{F (C}2_). _(1.10)

The following Theorem which appears as Theorem 2.2 in [21] gives the connection of the mapping F to the spectrum σ(C2_{) of the operator C}2_{= X}∗_X.

dynamical systems

Theorem 1.2.1. Let C and U be such that (1.10) holds. If eλ is an eigenvector

to C2 with eigenvalue λ, then C2(U∗eλ) = U∗(F (C2)eλ) = F (λ)(U∗eλ). Thus, the

vector U∗eλis either zero or is an eigenvector to C2with eigenvalue F (λ).

If X is invertible, that is, if U is unitary, then F (σ(C2₎₎_{⊆ σ(C}2_).

Using Theorem 1.2.1 above construction of representations X of (1.7) can be done as follows.

Take, if possible, a sequence of positive numbers{λk}k∈Zsuch that

F (λk) = λk+1

for all k∈ Z. Define for some orthonormal basis {ek}k∈N, the operators C and U in 2(Z) so that

C2ek= λkek, U ek= ek−1 (1.11) for all k∈ Z. Defined in this way, C and U satisfy (1.10) since U∗_(e_k_{) = e}_k+1_and

U C2_e k= λkek−1 F (C2)U ek= F (C2)ek−1= F (λk−1)ek−1= λkek−1 C2_U∗_e_k_{= λ}_k+1_e_k+1 U∗F (C2)ek= U∗F (λk)ek= λk+1U∗ek= λk+1ek+1

for all k∈ Z. Thus by (1.8), for any C and U satisfying (1.11) the operator X = UC

will be a representation of (1.7). It turns out that for some important important class of mappings, any irreducible representation X of (1.7) with invertible X is unitarily equivalent to a representation X of this form [21].

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 1.2.1. A point λ∈ C is said to be a periodic point of F if Fo(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

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and Operator Algebras

Definition 1.2.2. 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 A =    λ 0 . .. 0 Fo(n−1)(λ)    , B =      0 α 1 0 . .. ... 0 1 0      (1.12)

where |α| = 1 and λ ∈ P ern(F ), where P ern(F ) denotes the set of all periodic points of F with period n. 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 [21].

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 [Bs_At_{: B}u_Av_{] = 0 if and}

only if

Fo(k)(λ)tFo(k+n−u)(λ)v= Fo(k+s−u)(λ)vFo(k+n−u)(λ)t

for all k = 0, 1,· · · , n − 1, or equivalently

Fo(u)(λ)tµv= Fo(s)(µ)t, (1.13)

for all µ∈ Oλ.

Motivated by the results in [21], 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

Fo(u)(λ)tµv= Fo(s)(µ)t onC, where Fo(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.

Crossed product algebras and C

∗

−crossed products

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 ofC. 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 ofC.

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 and 4 in which we study commutants in crossed products of algebras of piecewise constant functions.

Consider a pair (A, φ) where A is an arbitrary associative commutative

com-plex algebra and φ :A → A is an automorphism. To this pair, we associate a

crossed product containing an isomorphic copy ofA, that is we define

A φZ := {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 defined by convolution twisted by the automorphism φ as follows;

(f g)(n) = k∈Z

f (k)φk(g(n− k)), (1.15)

f, g ∈ A φZ, n ∈ Z and φk denotes the k−fold composition of φ with itself.

A ×φZ is called the crossed product of A and Z under φ. A is called the coefficient algebra. A useful and convenient way of working withA φZ, is to write elements

f, g ∈ A φZ in the form f = n∈Zfnδn and g = n∈Zgmδm where fn =

f (n), gm= g(m) and δ = χ_{1}, where, for n, k∈ Z,

χ_{n}(k) =

1, if k = n

0, if k= n. (1.16)

It follows therefore that δn_{= χ} {n}.

Addition and scalar multiplication are then canonically defined and multipli-cation is determined by the relation

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and Operator Algebras

Definition 1.2.2. 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 A =    λ 0 . .. 0 Fo(n−1)(λ)    , B =      0 α 1 0 . .. ... 0 1 0      (1.12)

where |α| = 1 and λ ∈ P ern(F ), where P ern(F ) denotes the set of all periodic points of F with period n. 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 [21].

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 [Bs_At_{: B}u_Av_{] = 0 if and}

only if

Fo(k)(λ)tFo(k+n−u)(λ)v= Fo(k+s−u)(λ)vFo(k+n−u)(λ)t

for all k = 0, 1,· · · , n − 1, or equivalently

Fo(u)(λ)tµv= Fo(s)(µ)t, (1.13)

for all µ∈ Oλ.

Motivated by the results in [21], 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

Fo(u)(λ)tµv= Fo(s)(µ)t onC, where Fo(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.

Crossed product algebras and C

∗

−crossed products

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 ofC. 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 ofC.

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 and 4 in which we study commutants in crossed products of algebras of piecewise constant functions.

Consider a pair (A, φ) where A is an arbitrary associative commutative

com-plex algebra and φ :A → A is an automorphism. To this pair, we associate a

crossed product containing an isomorphic copy ofA, that is we define

A φZ := {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 defined by convolution twisted by the automorphism φ as follows;

(f g)(n) = k∈Z

f (k)φk(g(n− k)), (1.15)

f, g ∈ A φZ, n ∈ Z and φk denotes the k−fold composition of φ with itself.

A ×φZ is called the crossed product of A and Z under φ. A is called the coefficient algebra. A useful and convenient way of working withA φZ, is to write elements

f, g ∈ A φZ in the form f = n∈Zfnδn and g = n∈Zgmδm where fn =

f (n), gm= g(m) and δ = χ_{1}, where, for n, k∈ Z,

χ_{n}(k) =

1, if k = n

0, if k= n. (1.16)

It follows therefore that δn_{= χ} {n}.

Addition and scalar multiplication are then canonically defined and multipli-cation is determined by the relation

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and Operator Algebras

where m, n∈ Z and fn, gm∈ A.

Many studies have been carried out on A φZ for various coefficient alge-bras about maximal commutative subalgealge-bras, ideal intersections and many other properties in relation to the dynamics ofA [7, 12, 19, 26, 28, 29, 30]. One example

that has been studied in detail in 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 1.3.1. An algebraA 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) IfA is unital (that is, A has a ring identity 1A), then we assume that||1A|| = 1. Definition 1.3.2. A normed algebra which is complete in the metric induced by the norm is called a Banach algebra.

Definition 1.3.3. A Banach ∗−algebra is a complex Banach algebra A with a

conjugate linear involution ∗ (called the adjoint) which is an anti-isomorphism,

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 also called involutive algebras or

∗ − algebras. In a Banach ∗−algebra (or in any normed involutive algebra), A, ||a∗|| = ||a|| for every a ∈ A. (1.19) Therefore, from (1.18) and (1.19) we have that||a∗a|| ||a||2_{for every a}_{∈ A.}

Definition 1.3.4. AC∗−algebra A, is a Banach ∗−algebra with the additional

norm condition

||a∗a|| = ||a||2 _{for all a}_{∈ A} _(1.20)

(1.20) is called the C∗property.

Some examples of C∗− algebras include

Crossed product algebras and C

∗

−crossed products

• B(H), the algebra of linear operators on a Hilbert space H with pointwise

addition and scalar multiplication. Multiplication is defined by composition and the norm is given by

||T || := sup{||T (x)|| : ||x|| 1}.

• C(X) the algebra of continuous complex-valued functions on a compact

topological space Xwith the usual pointwise operations (addition, scalar multiplication and multiplication) and norm given by

||f|| := sup{|f(x)| : x ∈ X}.

As mentioned before, the class of C∗−algebras that has been extensively

stud-ied is the C∗_{− crossed product algebra. 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 supre-mum 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)φZ 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 φZ and n ∈ Z, where the bar denotes the usual pointwise complex conjugation. A = C(X) is a ∗−subalgebra of A φZ, namely as

{f : Z → A : f(n) = 0 if n = 0}.

The norm onA φZ is defined as follows; for f ∈ A φZ,

f =

n

fnδn where the fn∈ A and δ is unitary, that is, δ∗= δ−1,

||f|| =

n

||fn||∞.

CompletingA φZ in this norm yields a Banach ∗−algebra 1(Σ). That is

1(Σ) = f = n fnδn : n ||fn||∞<∞ ,

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and Operator Algebras

where m, n∈ Z and fn, gm∈ A.

Many studies have been carried out on A φZ for various coefficient alge-bras about maximal commutative subalgealge-bras, ideal intersections and many other properties in relation to the dynamics ofA [7, 12, 19, 26, 28, 29, 30]. One example

that has been studied in detail in 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 1.3.1. An algebraA 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) IfA is unital (that is, A has a ring identity 1A), then we assume that||1A|| = 1. Definition 1.3.2. A normed algebra which is complete in the metric induced by the norm is called a Banach algebra.

Definition 1.3.3. A Banach ∗−algebra is a complex Banach algebra A with a

conjugate linear involution ∗ (called the adjoint) which is an anti-isomorphism,

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 also called involutive algebras or

∗ − algebras. In a Banach ∗−algebra (or in any normed involutive algebra), A, ||a∗|| = ||a|| for every a ∈ A. (1.19) Therefore, from (1.18) and (1.19) we have that||a∗a|| ||a||2 _{for every a}_{∈ A.}

Definition 1.3.4. A C∗−algebra A, is a Banach ∗−algebra with the additional

norm condition

||a∗a|| = ||a||2_{for all a}_{∈ A} _(1.20)

(1.20) is called the C∗ property.

Some examples of C∗− algebras include

Crossed product algebras and C

∗

−crossed products

• B(H), the algebra of linear operators on a Hilbert space H with pointwise

addition and scalar multiplication. Multiplication is defined by composition and the norm is given by

||T || := sup{||T (x)|| : ||x|| 1}.

• C(X) the algebra of continuous complex-valued functions on a compact

topological space Xwith the usual pointwise operations (addition, scalar multiplication and multiplication) and norm given by

||f|| := sup{|f(x)| : x ∈ X}.

As mentioned before, the class of C∗−algebras that has been extensively

stud-ied is the C∗_{− crossed product algebra. 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 supre-mum 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)φZ 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 φZ and n ∈ Z, where the bar denotes the usual pointwise complex conjugation. A = C(X) is a ∗−subalgebra of A φZ, namely as

{f : Z → A : f(n) = 0 if n = 0}.

The norm onA φZ is defined as follows; for f ∈ A φZ,

f =

n

fnδn where the fn∈ A and δ is unitary, that is, δ∗= δ−1,

||f|| =

n

||fn||∞.

Completing A φZ in this norm yields a Banach ∗−algebra 1(Σ). That is

1(Σ) = f = n fnδn : n ||fn||∞ <∞ ,

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and Operator Algebras

with the operations ofA φZ 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 [32]. This C∗−algebra is usually

also denoted by C(X)φZ.

For a dynamical system Σ = (X, σ), a point x∈ X is called aperiodic if for

every nonzero n∈ Z, we have σn_(x)_{= x. 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 [28] states the equivalence between density of aperiodic points of X and maximal commutativity of C(X).

Theorem 1.3.1. The following properties are equivalent.

1. Σ is topologically free.

2. Every non-zero closed ideal I of C(X)φZ is such that I ∩ C(X) = {0}.

3. C(X) is a maximal abelian C∗−subalgebra of C(X) φZ.

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 [7, 11, 12, 19, 26, 28, 29, 30, 31].

In [26], a slightly different approach was taken. Connections between prop-erties of commutative subalgebras of crossed product algebras and propprop-erties of dynamical systems that are in many situations naturally associated with the con-struction were brought into an algebraic context. This consists of the concon-struction of a crossed product A φZ of an arbitrary subalgebra A of the algebra CX of functions on a set X (under the usual pointwise operations) byZ, where the latter acts onA 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 con-dition 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.

Crossed product algebras and C

∗

−crossed products

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 → σn_{(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 [26].

Definition 1.3.5. For any nonzero n∈ Z, we set

Sepn_A(X) = {x ∈ X | ∃h ∈ A : h(x) = ˜σn_(h)(x)_}, P er_An(X) = {x ∈ X | ∀h ∈ A : h(x) = ˜σn(h)(x)}, Sepn(X) = {x ∈ X | x = σn(x)}, P ern(X) = {x ∈ X | x = σn_(x)_}. Furthermore, let P er∞_A(X) = n∈Z\0 Sepn_A(X). P er∞(X) = n∈Z\0 Sepn(X). Finally, for f∈ A, set

supp(f ) ={x ∈ X | f(x) = 0}.

Then all these sets, except supp(f ) areZ−invariant and if A separates points of

X, for example, if A = C(X), the algebra of continuous complex-valued

func-tions on a compact Hausdorff topological space X, then Sepn_A(X) = Sepn(X) and

P ern

A(X) = P ern(X). Note also that SepnA(X) = X\ P ernA(X) and Sepn(X) =

X\ P ern_{(X). Furthermore Sep}n

A(X) = Sep−nA (X) with similar equalities for n and−n, (n ∈ Z) holding for P ern

A(X), Sepn(X) and P ern(X) as well. It should 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 [26] as Theorem 2.3.3 and Corollary 2.3.4 respectively.

Theorem 1.3.2. The unique maximal abelian subalgebra ofA σ˜Z that contains

A is precisely the set of elements A=

n∈Z

fnδn | for all n ∈ Z : fn|Sepn_A(X)≡ 0

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and Operator Algebras

with the operations of A φZ 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 [32]. This C∗−algebra is usually

also denoted by C(X)φZ.

For a dynamical system Σ = (X, σ), a point x ∈ X is called aperiodic if for

every nonzero n∈ Z, we have σn_(x)_{= x. 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 [28] states the equivalence between density of aperiodic points of X and maximal commutativity of C(X).

Theorem 1.3.1. The following properties are equivalent.

1. Σ is topologically free.

2. Every non-zero closed ideal I of C(X)φZ is such that I ∩ C(X) = {0}.

3. C(X) is a maximal abelian C∗−subalgebra of C(X) φZ.

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 [7, 11, 12, 19, 26, 28, 29, 30, 31].

In [26], a slightly different approach was taken. Connections between prop-erties of commutative subalgebras of crossed product algebras and propprop-erties of dynamical systems that are in many situations naturally associated with the con-struction were brought into an algebraic context. This consists of the concon-struction of a crossed product A φZ of an arbitrary subalgebra A of the algebra CX of functions on a set X (under the usual pointwise operations) byZ, 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 con-dition 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.

Crossed product algebras and C

∗

−crossed products

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 → σn_{(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 ˜σZ 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 [26].

Definition 1.3.5. For any nonzero n∈ Z, we set

Sepn_A(X) = {x ∈ X | ∃h ∈ A : h(x) = ˜σn_(h)(x)_}, P ern_A(X) = {x ∈ X | ∀h ∈ A : h(x) = ˜σn(h)(x)}, Sepn(X) = {x ∈ X | x = σn(x)}, P ern(X) = {x ∈ X | x = σn_(x)_}. Furthermore, let P er_A∞(X) = n∈Z\0 Sepn_A(X). P er∞(X) = n∈Z\0 Sepn(X). 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

func-tions on a compact Hausdorff topological space X, then Sepn_A(X) = Sepn(X) and

P ern

A(X) = P ern(X). Note also that SepnA(X) = X\ P ernA(X) and Sepn(X) =

X \ P ern_{(X). Furthermore Sep}n

A(X) = Sep−nA (X) with similar equalities for n and−n, (n ∈ Z) holding for P ern

A(X), Sepn(X) and P ern(X) as well. It should 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 [26] as Theorem 2.3.3 and Corollary 2.3.4 respectively.

Theorem 1.3.2. The unique maximal abelian subalgebra ofA σ˜Z that contains

A is precisely the set of elements A=

n∈Z

fnδn | for all n ∈ Z : fn|Sepn_A(X)≡ 0