Set theory and von Neumann algebras
... What is the motivation for giving these lectures, you ask. The answer is two-fold: On the one hand, there is a strong connection between (non-singular) group actions, countable Borel equivalence relations and von Neumann algebras, as we will see in Lecture 3 below. In the past decade, the knowledge ...
... What is the motivation for giving these lectures, you ask. The answer is two-fold: On the one hand, there is a strong connection between (non-singular) group actions, countable Borel equivalence relations and von Neumann algebras, as we will see in Lecture 3 below. In the past decade, the knowledge ...
Representation theory and applications in classical quantum
... Pv , it is said that the system is found in the state [v]. The number (p(v), p(w)) is interpreted as the probability to find the system in state [v], when measuring the observable Pv , when before measurement the system is known to be in state [w] (such a state could be prepared through measurement ...
... Pv , it is said that the system is found in the state [v]. The number (p(v), p(w)) is interpreted as the probability to find the system in state [v], when measuring the observable Pv , when before measurement the system is known to be in state [w] (such a state could be prepared through measurement ...
Introduction, Fields, Vector Spaces, Subspaces, Bases, Dimension
... proofs, as opposed to intuitive reasoning and sketches of proofs which are used in your lower division classes. You have probably seen rigorous proofs and the axiomatic method in a class in Euclidean geometry; we will be using this approach for linear algebra. The main proponents of this approach we ...
... proofs, as opposed to intuitive reasoning and sketches of proofs which are used in your lower division classes. You have probably seen rigorous proofs and the axiomatic method in a class in Euclidean geometry; we will be using this approach for linear algebra. The main proponents of this approach we ...
Ch-3 Vector Spaces and Subspaces-1-web
... (which is the only vector required to be in all vector spaces). Thus V = (V, K, +, , 0 ). (Since we use juxtaposition to indicate scalar multiplication, we have no symbol for multiplication of a vector by a scalar and hence just leave a space.) We consider some properties that hold in all vector spa ...
... (which is the only vector required to be in all vector spaces). Thus V = (V, K, +, , 0 ). (Since we use juxtaposition to indicate scalar multiplication, we have no symbol for multiplication of a vector by a scalar and hence just leave a space.) We consider some properties that hold in all vector spa ...
An introduction to some aspects of functional analysis, 5: Smooth
... but finitely many r, in which case it can be used to define seminorms as in the previous section. One can get the same topologies on C(U ), C l (U ) for each k ∈ Z+ , and C ∞ (U ) using the seminorms associated to these compact sets Ar when Ar 6= ∅, and there are only countably many of these seminor ...
... but finitely many r, in which case it can be used to define seminorms as in the previous section. One can get the same topologies on C(U ), C l (U ) for each k ∈ Z+ , and C ∞ (U ) using the seminorms associated to these compact sets Ar when Ar 6= ∅, and there are only countably many of these seminor ...
Explicit product ensembles for separable quantum states
... For more than one qubit, the possibilities for discrete representations that follow from the continuous representation are even more diverse. The simplest possibility is to use the same set of pure-state projectors for each qubit, but this is not necessary. One can use a different set of projectors ...
... For more than one qubit, the possibilities for discrete representations that follow from the continuous representation are even more diverse. The simplest possibility is to use the same set of pure-state projectors for each qubit, but this is not necessary. One can use a different set of projectors ...
3 Lecture 3: Spectral spaces and constructible sets
... (4) If X is qcqs then the constructible subsets of X are exactly the finite Boolean combinations of quasi-compact open subsets of X (so one doesn’t need to speak of retrocompactness in such cases). This is the typical situation of interest. (Recall the exercise from Hartshorne’s textbook on algebra ...
... (4) If X is qcqs then the constructible subsets of X are exactly the finite Boolean combinations of quasi-compact open subsets of X (so one doesn’t need to speak of retrocompactness in such cases). This is the typical situation of interest. (Recall the exercise from Hartshorne’s textbook on algebra ...
Orthogonal Transformations and Matrices
... G. Prove that the rows of an orthogonal matrix are also orthonormal. [Hint: don’t forget that the rows of A are the columns of AT . Remember that the inverse of an orthogonal map is also orthogonal.] Solution note: Say A is orthogonal. Then the map TA is orthogonal. Hence its inverse is orthogonal, ...
... G. Prove that the rows of an orthogonal matrix are also orthonormal. [Hint: don’t forget that the rows of A are the columns of AT . Remember that the inverse of an orthogonal map is also orthogonal.] Solution note: Say A is orthogonal. Then the map TA is orthogonal. Hence its inverse is orthogonal, ...
Noncommutative Lp-spaces of W*-categories and their applications
... We consider vector bundles equipped with a more general type of inner product with values in Dens2d (X) for some d ∈ R. Here it is essential that d is real because we need Dens+ 2d (X) for the positivity property. Such an inner product equips every fiber Vx of V with an inner product with values in ...
... We consider vector bundles equipped with a more general type of inner product with values in Dens2d (X) for some d ∈ R. Here it is essential that d is real because we need Dens+ 2d (X) for the positivity property. Such an inner product equips every fiber Vx of V with an inner product with values in ...
Mathematical structures
... is relevant when we discuss notions such as distance and continuity of functions. We focus mainly on metric spaces and we prove for instance Banach’s fixed point theorem and Baire’s category theorem. After some preparation we also give the basic existence theorem for variational problems, i.e. that ...
... is relevant when we discuss notions such as distance and continuity of functions. We focus mainly on metric spaces and we prove for instance Banach’s fixed point theorem and Baire’s category theorem. After some preparation we also give the basic existence theorem for variational problems, i.e. that ...
Chapter 3 Representations of Groups
... Proof: T h e image L = A X of X under A is a subspace of Y. L is invariant under S(g) because S ( g ) A z = A ( T ( g ) ) x = Ax' E L. S is irreducible and L is therefore the nullspace or the whole space Y. In the first case A = 0, in the second case we find that A is surjective and we show that A i ...
... Proof: T h e image L = A X of X under A is a subspace of Y. L is invariant under S(g) because S ( g ) A z = A ( T ( g ) ) x = Ax' E L. S is irreducible and L is therefore the nullspace or the whole space Y. In the first case A = 0, in the second case we find that A is surjective and we show that A i ...
Fell bundles associated to groupoid morphisms
... We first discuss a class of examples considered by Fell. Let G be a discrete group, let K be a normal subgroup and let H = G/K with π : G → H the canonical morphism. Then we get a Fell bundle E over H with the fiber Cr∗ (K) over the identity element, such that Cr∗ (G) ∼ = Cr∗ (E). It is equivalent t ...
... We first discuss a class of examples considered by Fell. Let G be a discrete group, let K be a normal subgroup and let H = G/K with π : G → H the canonical morphism. Then we get a Fell bundle E over H with the fiber Cr∗ (K) over the identity element, such that Cr∗ (G) ∼ = Cr∗ (E). It is equivalent t ...
The Chiral Ring
... so we see that the ground states provide a realization of the chiral ring. However, we do not always get a one-to-one correspondence (e.g. topological LandauGinzburg models with A-twist), but for many theories it is. Especially for the Sigma model with A-twist, which is the theory we are interested ...
... so we see that the ground states provide a realization of the chiral ring. However, we do not always get a one-to-one correspondence (e.g. topological LandauGinzburg models with A-twist), but for many theories it is. Especially for the Sigma model with A-twist, which is the theory we are interested ...
Representations of Locally Compact Groups
... a more concrete group or algebra consisting of matrices or operators. In this way we can study an algebraic object as collection of symmetries of a vector space. Hence we can apply the methods of linear algebra and functional analysis to the study of groups and algebras. Representation theory also p ...
... a more concrete group or algebra consisting of matrices or operators. In this way we can study an algebraic object as collection of symmetries of a vector space. Hence we can apply the methods of linear algebra and functional analysis to the study of groups and algebras. Representation theory also p ...
Markov property in non-commutative probability
... of classical probability theory was subsumed into classical measure theory by A.N. Kolmogorov [34], the quantum or non-commutative probability theory was induced by the quantum theory and was incorporated into the beginnings of non-commutative measure theory by J. von Neumann [48]. In this concept, ...
... of classical probability theory was subsumed into classical measure theory by A.N. Kolmogorov [34], the quantum or non-commutative probability theory was induced by the quantum theory and was incorporated into the beginnings of non-commutative measure theory by J. von Neumann [48]. In this concept, ...
L17-20
... 2. The Kraus representation theorem says that any quantum operation can be realized by a measurement model in which the ancilla’s initial state is pure. It is clear why we need only consider initial pure states for the ancilla: if we find a measurement model with the ancilla initially in a mixed sta ...
... 2. The Kraus representation theorem says that any quantum operation can be realized by a measurement model in which the ancilla’s initial state is pure. It is clear why we need only consider initial pure states for the ancilla: if we find a measurement model with the ancilla initially in a mixed sta ...
the structure of certain operator algebras
... of A are finite-dimensional and of the same degree). In Theorem 5.1 we show that the structure space of a CCi?-algebra, while not necessarily Hausdorff, is at least of the second category. This information is used to make two further advances; first to the case where all A/P are finite-dimensional, ...
... of A are finite-dimensional and of the same degree). In Theorem 5.1 we show that the structure space of a CCi?-algebra, while not necessarily Hausdorff, is at least of the second category. This information is used to make two further advances; first to the case where all A/P are finite-dimensional, ...
Irreducible Tensor Operators and the Wigner
... of the Hamiltonian are discussed in Sec. 15 below, where it is shown that the energy eigenspaces consist of one or more irreducible subspaces under rotations. In fact, apart from exceptional cases like the electrostatic model of hydrogen, each energy eigenspace consists of precisely one irreducible ...
... of the Hamiltonian are discussed in Sec. 15 below, where it is shown that the energy eigenspaces consist of one or more irreducible subspaces under rotations. In fact, apart from exceptional cases like the electrostatic model of hydrogen, each energy eigenspace consists of precisely one irreducible ...
Milan Merkle TOPICS IN WEAK CONVERGENCE OF PROBABILITY
... sequence Xn converges weakly to X and write Xn =⇒ X if and only if Pn =⇒ P . As we shall see in Section 4, there are many stronger convergence concepts than the introduced one. However, the weak convergence is a very powerful tool in Probability Theory, partly due to its comparative simplicity and p ...
... sequence Xn converges weakly to X and write Xn =⇒ X if and only if Pn =⇒ P . As we shall see in Section 4, there are many stronger convergence concepts than the introduced one. However, the weak convergence is a very powerful tool in Probability Theory, partly due to its comparative simplicity and p ...
NATURAL EXAMPLES OF VALDIVIA COMPACT SPACES 1
... In Section 3 we investigate linearly ordered compact spaces. We present a conjecture on a characterization of Valdivia compacta among linearly ordered ones and we show that it is true for scattered or connected spaces. We also show that linearly ordered Valdivia compact spaces are hereditarily Valdi ...
... In Section 3 we investigate linearly ordered compact spaces. We present a conjecture on a characterization of Valdivia compacta among linearly ordered ones and we show that it is true for scattered or connected spaces. We also show that linearly ordered Valdivia compact spaces are hereditarily Valdi ...
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of ""dropping the altitude"" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.