Quantum tomography is a method of reconstructing a quantum state based on information about the probability distributions associated with the quadrature operators, linear combinations of position...Show moreQuantum tomography is a method of reconstructing a quantum state based on information about the probability distributions associated with the quadrature operators, linear combinations of position and momentum. The theory says that if one knows these distributions precisely, the quantum state can be faithfully reconstructed. This thesis delves into the mathematical formalism behind quantum tomography, how it can be applied to quantum systems one might encounter in practice. We rigorously define the quantum tomogram of a state and give an example of an experimental set-up which allows one to measure the quadratures of single mode light.Show less
Let G be a locally compact Hausdorff group and let ν be a fixed Haar measure on G. If µ is a bounded positive Radon measure on G and f ∈ L p (G, ν), their convolution product µ ∗ f exists. Thus...Show moreLet G be a locally compact Hausdorff group and let ν be a fixed Haar measure on G. If µ is a bounded positive Radon measure on G and f ∈ L p (G, ν), their convolution product µ ∗ f exists. Thus every bounded Radon measure acts on L p (G, ν) as a convolution operator. It is easy to see that the action of a bounded Radon measure on L p (G, ν) gives rise to an injective Banach algebra homomorphism from the Banach algebra of bounded Radon measures on G to the Banach algebra of regular operators from L p (G, ν) to L p (G, ν). We will investigate when this algebra homomorphism is also a (isometric) lattice isomorphism. For each a ∈ G we have a right translation operator ρa on L p (G, ν). The regular commutant of the right translation operators is the set of regular operators T from L p (G, ν) to L p (G, ν) such that ρaT = T ρa for all a ∈ G. We will show that the regular commutant of the right translation operators is also a Banach lattice. The bounded Radon measures on G also form a Banach lattice. It is then investigated when the injective Banach algebra homomorphism is actually a lattice isomorphism from the bounded Radon measures on G onto the regular commutant of the right translation operators. This is the main result of this thesis. We start by showing that each positive regular operator in the regular commutant of the right translation operators is in fact a convolution operator related to a positive Radon measure. For p = 1, we improve the result by showing that the Radon measure must be bounded and we show that our algebra isomorphism is then an isometric lattice isomorphism. For p ∈ (1, ∞) the same results follow under the extra assumption that G is also amenable.Show less
We first introduce the concept of partner potentials in non-relativistic quantum mechanics, i.e. a pair of potentials with the same spectrum, possibly except for a zero-energy ground state. We use...Show moreWe first introduce the concept of partner potentials in non-relativistic quantum mechanics, i.e. a pair of potentials with the same spectrum, possibly except for a zero-energy ground state. We use this to define a family of partner potentials, giving us a technique to calculate the entire spectrum of a potential. The mechanism of partner potentials is then used for a quantum mechanical model of supersymmetry. It turns out that a special class of potentials exists where the spectrum can be determined very quickly using the techniques developed. We explore some of these potentials, called shape invariant potentials or SIPs and discover some simple properties of them. Finally, we take a quick look at a Hamiltonian with a p 4 term in it, discovering that for a small class of potentials, we can make a supersymmetric quantum mechanical model out of it.Show less
If G is a locally compact group, then L 1 (G) acts by convolution on L p (G). This action yields a homomorphism of the Banach algebra L 1 (G) into the bounded operators on L p (G). For finite p it...Show moreIf G is a locally compact group, then L 1 (G) acts by convolution on L p (G). This action yields a homomorphism of the Banach algebra L 1 (G) into the bounded operators on L p (G). For finite p it is known, with the help of rather complicated tools, that this action gives a lattice homomorphism of L 1 (G) into the regular operators on L p (G). In this master’s thesis we generalize this result by letting L 1 (G) act on so-called translation invariant Banach function spaces on G, or, more precisely, on the largest subspaces of such spaces where this action can be meaningfully defined. We show, using methods that are considerably simpler than those used in the previous proof, that under mild assumptions this action is also a lattice homomorphism.Show less
In deze scriptie wordt een stelling behandeld van de wiskundige John W. Lamperti [1]. De stelling laat op een verrassende manier zien hoe isometrie¨en tussen L p -ruimten zich laten beschrijven....Show moreIn deze scriptie wordt een stelling behandeld van de wiskundige John W. Lamperti [1]. De stelling laat op een verrassende manier zien hoe isometrie¨en tussen L p -ruimten zich laten beschrijven. Hierbij spelen bepaalde afbeeldingen, de reguliere verzamelingsisomorfismen, een belangrijke rol. De stelling zegt dat isometrie¨en in de zogenaamde Lampertivorm te schrijven zijn als 1 ≤ p < ∞ en p 6= 2. Het zal blijken dat voor het geval p = 2 isometrie¨en bestaan die zich niet zo makkelijk laten beschrijven. Dit is intu¨ıtief in te zien door isometrie¨en op (R 2 , || · ||p) te beschouwen. Een isometrie moet de eenheidscirkel voor de p-norm op zichzelf afbeelden. Voor p > 2 en 1 ≤ p < 2 is de eenheidscirkel vierkantvormig en bestaan beperkte mogelijkheden voor een symmetrie en dus ook voor een isometrie. Als p = 2, dan is de verzameling symmetrie¨en overaftelbaar en zal een isometrie dus niet altijd dezelfde eigenschappen hebben als wanneer p 6= 2. Deze scriptie is bedoeld voor wiskundigen die bekend zijn met maattheorie. De bronnen waarop deze scriptie gebaseerd is, zijn voornamelijk [1] en [2]. Bijna alle idee¨en zijn afkomstig van deze bronnen, echter niet alle uitwerkingen.Show less
In this thesis we characterize two types of operators between C(K)-spaces: lattice and algebra homomorphisms. This characterization is nice, because something that seems to be rather big and...Show moreIn this thesis we characterize two types of operators between C(K)-spaces: lattice and algebra homomorphisms. This characterization is nice, because something that seems to be rather big and incomprehensible, suddenly becomes simple and comprehensible. With C(K) is meant the space of all continuous functions on a compact Hausdorff space K. One can give these spaces a special structure, namely the structure of a lattice or an algebra. In this thesis we consider the morphisms between C(K)-spaces that respect these structures. C(K)-spaces are not simply lattices and algebras. Because of the completeness with respect to the norm ||f|| = sup{|f(x)| : x ∈ K}, these spaces are also Banach lattices and Banach algebras. We treat this in the second section. We also give some more examples of Banach lattices and Banach algebras. To prove the characterization theorems, we need some useful results, like The Riesz Representation Theorem. We discuss these results in Section 3. Before we formulate and prove the characterization theorems in Section 5, in Section 4 we give the proof of some lemmas that are essential in the proof. Finally, we consider the lattice and algebra automorphisms. The structure of the groups of such automorphisms can be described completely, and this is done in the final section. At the end of this introduction, I would like to thank my advisor M.F.E. de Jeu for his kind help and the friendly contact during my project.Show less
Is there a natural one-to-one correspondence between positive representations of spaces of continuous functions and positive spectral measures on Banach lattices? That is the question that we have...Show moreIs there a natural one-to-one correspondence between positive representations of spaces of continuous functions and positive spectral measures on Banach lattices? That is the question that we have investigated in this thesis. Representations and spectral measures are key notions throughout all sections. After having given precise definitions of a positive spectral measure, a unital positive spectral measure, a positive representation, and a unital positive representation, we prove that there is a one-to-one relationship between the positive spectral measures and positive representations, and the unital positive spectral measures and unital positive representations, respectively. For example, between a unital positive representation ρ : C(K) → Lr(X) and a unital positive spectral measure E : Ω → Lr(X), where K is a compact Hausdorff space, Ω ⊆ P(K) the Borel σ-algebra, X a reflexive Banach lattice, and Lr(X) the space of all regular operators on X. The map ρ : C(K) → Lr(C(K)), where K is a compact Hausdorff space, defined as pointwise multiplication on C(K), is a unital positive representation. The subspace ρ[f] := ρ(C(K))f ⊆ C(K), for an f ∈ C(K), has two partial orderings. One is inherited from C(K), the other is newly defined. Using the defining property of elements of ρ[f], we prove that they are equal. A similar correspondence between representations and spectral measures on a general Banach space only exists under the assumption of R-boundedness ([PR07]). We show that the spectral measure that is generated via that correspondence is equal to the unital positive spectral measure that is generated from a unital positive representation in our setting. Finally, we show that the correspondence between our unital positive spectral measures and unital positive representations is relevant in the context of covariance, belonging to the theory of crossed products, as well. We look at an example in which there is a one-to-one relationship between covariant representions and covariant spectral measures.Show less
In this thesis we will get familiar with the Theorem of Banach and Stone and a nice application. The Theorem is a classic result within the theory of spaces of continuous maps on compact Hausdorff...Show moreIn this thesis we will get familiar with the Theorem of Banach and Stone and a nice application. The Theorem is a classic result within the theory of spaces of continuous maps on compact Hausdorff spaces and is named after the mathematicians Stefan Banach and Marshall Stone. Let C(K) denote the Banach space consisting of all real- or complex-valued continuous functions on a compact Hausdorff space K, with the supremumnorm kfk∞ = sup{|f(x)| : x ∈ K}. Stefan Banach thought about the case when there is a isometric function between two such spaces. In 1932 he solved his problem for compact metric spaces K by describing that isometric map. This was extended by Marshall Stone in 1937 to general compact Hausdorff spaces K. We will consider this generalized version in this paper. The contents of this paper are as follows. In the next Section, we will discuss some basic results from different fields in mathematics. We will conclude this Section with a precise formulation of the Banach-Stone Theorem, Theorem 2.14. In Section 3 we consider more advanced results from linear functional analysis. In Section 4 we prove some important theorems from topology which are essential in the proof of the Banach-Stone Theorem. Then we have enough knowledge to prove it in Section 5. We conclude this thesis with Section 6, where we apply the Banach-Stone Theorem to obtain a description of the structure on the group of isometric isomorphisms from C(K) to C(K). Prior to reading this thesis, one should have some basic knowledge of topology and linear functional analysis. A short recap is given, but for details one could consult [6] or [7]. Before we take off, I would like to express my appreciation to my advisor M.F.E. de Jeu for the support and interesting conversations during this project.Show less
It this thesis we investigate whether an analogue of the von Neumann Bicommutant Theorem and related results are valid for Riesz spaces. Let H be a Hilbert space and D ⊂ Lb(H) a ∗-invariant subset....Show moreIt this thesis we investigate whether an analogue of the von Neumann Bicommutant Theorem and related results are valid for Riesz spaces. Let H be a Hilbert space and D ⊂ Lb(H) a ∗-invariant subset. The bicommutant D00 equals P(D0 ) 0 , where P(D0 ) denotes the set of projections in D0 . Since the sets D00 and P(D0 ) 0 agree, there are multiple possibilities to define an analogue of bicommutant for Riesz spaces. Let E be a Dedekind complete Riesz space and A ⊂ Ln(E) a subset. Since the band generated by the projections in Ln(E) is given by Orth(E) and order projections in the commutant correspond bijectively to reducing bands, our approach is to define the bicommutant of A on E by U := (A 0 ∩ Orth(E))0 . Our first result is that the bicommutant U equals {T ∈ Ln(E) : T is reduced by every A - reducing band}. Hence U is fully characterized by its reducing bands. This is the analogue of the fact that each von Neumann algebra in Lb(H) is reflexive. This result is based on the following two observations. Firstly, in Riesz spaces there is a one-to-one correspondence between bands and order projections, instead of a one-to-one correspondence between closed subspaces and projections. Secondly, every ∗-invariant subset of Lb(H) is reduced by each invariant subspace. Therefore, “invariant closed subspaces” is replaced by “reducing bands” in the reflexivity result. Similarly, we obtain Schur’s Lemma with “invariant subspaces” replaced by “reducing bands”. There is no natural counterpart of the adjoint for Riesz spaces. However, we may define A ⊂ Ln(E) to have the ∗-property, if every A -invariant band is reducing. If A has the ∗-property, we obtain our classical reflexivity result and Schur’s Lemma as known for Hilbert spaces. An instance in which A has the ∗-property is a subgroup A of the Riesz automorphisms on E. Furthermore, we obtain that the bicommutant U is a unital band algebra. Conversely, if A is a unital band algebra with the ∗-property and x ∈ E, then, for every operator T ∈ U , the element T x is approached in order by a net from A x. If E is atomic, we get approximation in order of each T ∈ U by a net of operators from A . Therefore, if E is an atomic Dedekind complete Riesz space and A ⊂ Ln(E) has the ∗-property, then A equals its bicommutant (A 0 ∩ Orth(E))0 if and only if A is a unital band algebra. So we retrieve an analogue of the von Neumann Bicommutant Theorem for atomic Riesz spaces. A direct consequence of the von Neumann Bicommutant Theorem is that each von Neumann algebra is the commutant of a group of unitaries. Similarly, the order bicommutant (A 0 ∩Orth(E))0 is the commutant of some group of invertible orthomorphisms for every A ⊂ Ln(E). Combining these facts gives that, if E is atomic, then each A ⊂ Ln(E) with the ∗-property is a unital band algebra if and only if A is the commutant of some group of invertible orthomorphisms. To obtain the above results, we study operator algebras on Riesz spaces, a subject which is hardly treated in literature at the moment. Moreover, we deal with invariance questions under a set of operators on a Riesz space.Show less
This thesis is about amenability. There are several contexts in which this property can be used and we will discuss what these are and how they relate to one another and, simultaneously, give an...Show moreThis thesis is about amenability. There are several contexts in which this property can be used and we will discuss what these are and how they relate to one another and, simultaneously, give an overview of what is done in this thesis. The notion of amenable groups arose in the first half of the 20th century in the context of the famous BanachTarski paradox:Show less
In this thesis finite dimensional Riesz spaces will be studied. Riesz spaces are real vector spaces equipped with a partial order. Under this partial order the Riesz space must satisfy some axioms,...Show moreIn this thesis finite dimensional Riesz spaces will be studied. Riesz spaces are real vector spaces equipped with a partial order. Under this partial order the Riesz space must satisfy some axioms, including the axiom that it is a lattice. In the first chapter the concept of a lattice and a Riesz space will be introduced and some examples will be given. Also some general definitions and properties will be given. Furthermore two partial orders on R n, the pointwise and the lexicographical order, will be introduced. In the second chapter homomorphisms between Riesz spaces will be discussed. In chapter three we will use these homomorphisms to understand the structure of finite dimensional Riesz spaces. Here we will prove that each finite dimensional Riesz space E can be built up as the direct sum Ln j=1 Ij of a finite number of Riesz spaces, where every Ij can be constructed out of a lower dimensional Riesz space, using a variant of the lexicographical order. In the last chapter we will use these results to find the automorphism groups of finite dimensional Riesz spaces.Show less
In elementary functional analysis, one is taught that all normed vector spaces of given finite dimension over the reals or the complex numbers are isomorphic. One therefore could conclude that the...Show moreIn elementary functional analysis, one is taught that all normed vector spaces of given finite dimension over the reals or the complex numbers are isomorphic. One therefore could conclude that the theory of finite-dimensional vector spaces is trivial and not worth investigating. In this bachelor thesis, I will try to convince the reader that this is not the case. For instance, when investigating general Banach spaces one comes upon the field of local theory. This research area studies finite-dimensional normed spaces and the relation between a Banach space and its finite-dimensional subspaces. We will delve into a small part of this theory by introducing the BanachMazur distance, a fundamental tool when studying finite-dimensional normed spaces. Methods developed in the study of this distance have helped answer several longstanding questions about Banach spaces, and are used in other areas such as operator theory and harmonic analysis [Tomczak-Jaegermann 1989, p. x,xi]. The Banach-Mazur distance defines a metric on a set of equivalence classes of normed spaces, thereby defining the ’space of spaces’ from the title of this thesis. We can then ask ourselves several questions about this space, amongst which whether the space is bounded and compact. The first questions leads to the theorem of John (1948), which gives an upper bound for the diameter of this space. The problem of determining the quality of this upper bound was left open for some time, until it was solved in 1981 by Gluskin. His proof uses an interesting measure-theoretic approach which has since then been used in several other results. This theorem is the main piece of this thesis, and we will work out in more detail his short but very technical proof [Gluskin 1981]. In the paragraph following this introduction, we introduce the Banach-Mazur distance as a way of measuring how different two norms on isomorphic vector spaces are. We explore some of the properties of this distance. Using this distance, we can define a metric on the set of equivalence classes of n-dimensional (n ∈ N) normed spaces over the reals, and in the third section we go on to explore some of the properties of this metric space. In the fourth paragraph, we introduce the theorem of Gluskin to which we alluded previously, and go on to prove it, using the results from two appendices. We conclude with a paragraph stating some additional results using the same technique as Gluskin’s proofShow less
The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra...Show moreThe study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. This particular paper focuses on Gelfand theory — the relation between multiplicative linear functionals on a commutative Banach algebra and its maximal ideals, as well as with the spectra of its elements. Most of the content of chapters 1 thorough 3 is meant, in one way or another, to lead towards this theory. The central ingredient of Gelfand theory is the well-known Gelfand-Mazur theorem which says that if a Banach algebra is a division algebra then it is isomorphic to C. The first chapter is a purely algebraic one and provides us with all the necessary algebraic techniques, particularly concerning algebras without identity. The second and third chapters introduce normed algebras and Banach algebra and other concepts like the spectrum, and prove several important results among which the Gelfand-Mazur theorem. The fourth chapter is the pivotal one — where Gelfand theory is developed. In the fifth chapter several examples of Banach algebras are discussed in detail, together with their Gelfand representations. Some practical applications of the theory are also mentioned, among which Wiener’s famous theorem about zeroes of functions with absolutely Fourier series, proven entirely from the context of Banach algebras.Show less
