Showing papers on "Operator algebra published in 2003"

Geometric Algebra for Physicists

Abstract: Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.

Introduction to Operator Space Theory

Abstract: The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.

Non-commutative space-time of Doubly Special Relativity theories

Abstract: Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy–momentum sector, each of whose can be promoted to the κ-Poincare quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space–time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space–time of the DSR theory is unique and related to the theory with non-commutative space–time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space–time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space–time, its intrinsic length par...

Polynomial invariants of four qubits

Abstract: We describe explicitly the algebra of polynomial functions on the Hilbert space of four-qubit states that are invariant under the group SL(2,C){sup 4} of stochastic local quantum operations assisted by classical communication. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.

Non(anti)commutative superspace

Abstract: We investigate the most general non(anti)commutative geometry in N = 1 four-dimensional superspace, invariant under the classical (i.e., undeformed) supertranslation group. We find that a nontrivial non(anti)commutative superspace geometry compatible with supertranslations exists with non(anti)commutation parameters which may depend on the spinorial coordinates. The algebra is in general nonassociative. Imposing associativity introduces additional constraints which, however, allow for nontrivial commutation relations involving fermionic coordinates. We obtain explicitly the first three terms of a series expansion in the deformation parameter for a possible associative -product. We also consider the case of N = 2 Euclidean superspace where the different conjugation relations among spinorial coordinates allow for a more general supergeometry.

Geometry of state spaces of operator algebras

Abstract: Preface * JB-Algebras * JBW-Algebras * Structure of JBW-Algebras * Representations of JB-Algebras * State Spaces of Jordan Algebras * Dynamical Correspondences * General Compressions * Spectral Theory * Characterization of Jordan Algebra State Spaces * Characterization of Normal State Spaces of von Neumann Algebras * Characterization of C*-Algebra State Spaces * Appendix * Bibliography * Index

UV divergence-free QFT on noncommutative plane

Abstract: We formulate noncommutative qauntum field theory in terms of fields defined as mean value over coherent states of the noncommutative plane. No -product is needed in this formulation and noncommutativity is carried by a modified Fourier transform of fields. As a result the theory is UV finite and the cutoff is provided by the noncommutative parameter θ.

Rationality, regularity, and ₂-cofiniteness

Abstract: We demonstrate that, for vertex operator algebras of CFT type, C 2 -cofiniteness and rationality is equivalent to regularity. For C 2 -cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and C 2 -cofinite, V + L is C 2 -cofinite, and the fusion rules are finite.

Quantum Measure Theory

Abstract: Preface. 1: Introduction. 2: Operator Algebras. 2.1. C*-Algebras. 2.2. Von Neumann Algebras. 2.3. Jordan Algebras And Ordered Structures. 3: Gleason Theorem. 3.1. Reduction To Three-Dimensional Space. 3.2. Regularity Of Frame Functions On R3. 3.3. Boundedness Of Frame Functions. 3.4. Historical Remarks And Comments. 4: Completeness Criteria. 4.1. Functiona1 Completeness Criteria. 4.2. Algebraic Completeness Criteria. 4.3. Measure Theoretic Completeness Criteria. 4.4. Historical Remarks And Comments. 5: Generalized Gleason Theorem. 5.1. The Mackey-Gleason Problem. 5.2. Reduction To Scalar Quasi-Functionals. 5.3. Linear Extensions Of Measures On Type In Algebras. 5.4. Linear Extensions Of Measures On Infinite Algebras. 5.5. Linear Extensions Of Measures On Finite Algebras. 5.6. Historical Remarks And Comments. 6: Basic Principles Of Quantum Measure Theory. 6.1. Boundedness Of Completely Additive Measures. 6.2. Yosida-Hewitt Decompositions Of Quantum Measures. 6.3. Convergence Theorems. 6.4. Historical Remarks And Comments. 7: Applications Of Gleason Theorem. 7.1. Multiform Gleason Theorem And Decoherence. 7.2. Velocity Maps And Derivations. 7.3. Approximate Hidden Variables. 7.4. Historical Remarks And Comments. 8: Orthomorphisms Of Projections. 8.1. Orthomorphisms Of Projection Lattices. 8.2. Countable Additivity Of *-Homomorphisms. 8.3. Historical Remarks And Comments. 9: Restrictions And Extensions Of States. 9.1. Restriction Properties Of Pure States. 9.2. Gleason Type Theorems For Quantum Logics. 9.3. Historical Remarks And Comments. 10: Jauch-Piron States. 10.1. Basic Properties Of Jauch States. 10.2. Nonsingularity Of Jauch-Piron States. 10.3. Countable Additivity Of States. 10.4. Historical Remarks And Comments. 11: Independence Of Quantum Systems. 11.1. Independence In Classical And Quantum Theory. 11.2. Independence Of C*-Algebras. 11.3. Independence Of Von Neumann Algebras. 11.4. Historical Remarks And Comments. Bibliography. Index.

Conformal subnets and intermediate subfactors

Abstract: Given an irreducible local conformal net 𝒜 of von Neumann algebras on S 1 and a finite-index conformal subnet ℬ⊂𝒜, we show that 𝒜 is completely rational iff ℬ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [27] hold. Our proofs are based on an analysis of the net inclusion ℬ⊂𝒜; among other things we show that, for a fixed interval I, every von Neumann algebra  intermediate between ℬ(I) and 𝒜(I) comes from an intermediate conformal net ℒ between ℬ and 𝒜 with ℒ(I)=. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set ℑ(𝒩,ℳ) of intermediate subfactors in an irreducible inclusion of factors 𝒩⊂ℳ with finite Jones index [ℳ:𝒩]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of ℑ(𝒩,ℳ) which depends only on [ℳ:𝒩].

Rationality, Quasirationality and Finite W-Algebras

Abstract: Some of the consequences that follow from the C 2 condition of Zhu are analysed. In particular it is shown that every conformal field theory satisfying the C 2 condition has only finitely many n-point functions, and this result is used to prove a version of a conjecture of Nahm, namely that every representation of such a conformal field theory is quasirational. We also show that every such vertex operator algebra is a finite W-algebra, and we give a direct proof of the convergence of its characters as well as the finiteness of the fusion rules.

Real Operator Algebras

Abstract: Real Banach and Hilbert Spaces Real Banach Algebras Real Banach * Algebras Fundamentals of Real Von Neumann Algebras Fundamentals of Real C*-Algebras Real W*-Algebras Gelfand-Naimark Conjecture in the Real Case Classification of Real W*-Algebras Real Reduction Theory (AF) Real C*-Algebras.

The Fuzzy Ginsparg-Wilson Algebra: A Solution of the Fermion Doubling Problem

Abstract: The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)A anomaly and index theory correctly.

Partial Dynamical Systems and the KMS Condition

Abstract: Given a countably infinite 0–1 matrix A without identically zero rows, let 𝒪 A be the Cuntz–Krieger algebra recently introduced by the authors and 𝒯 A be the Toeplitz extension of 𝒪 A , once the latter is seen as a Cuntz–Pimsner algebra, as recently shown by Szymanski. We study the KMS equilibrium states of C * -dynamical systems based on 𝒪 A and 𝒯 A , with dynamics satisfying for the canonical generating partial isometries s x and arbitrary real numbers N x > 1. The KMSβ states on both 𝒪 A and 𝒯 A are completely characterized for certain values of the inverse temperature β, according to the position of β relative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for 𝒪 A are derived from those for 𝒯 A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for 𝒪 n and of Enomoto, Fujii and Watatani for 𝒪 A as particular cases.

(0,2) Duality

Abstract: We construct dual descriptions of (0, 2) gauged linear sigma models. In some cases, the dual is a (0, 2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0, 2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0, 2) generalization of the quantum cohomology ring of (2, 2) theories.

Systematics of Quarter BPS operators in N=4 SYM

Abstract: A systematic construction is presented of 1/4 BPS operators in N=4 superconformal Yang-Mills theory, using either analytic superspace methods or components. In the construction, the operators of the classical theory annihilated by 4 out of 16 supercharges are arranged into two types. The first type consists of those operators that contain 1/4 BPS operators in the full quantum theory. The second type consists of descendants of operators in long unprotected multiplets which develop anomalous dimensions in the quantum theory. The 1/4 BPS operators of the quantum theory are defined to be orthogonal to all the descendant operators with the same classical quantum numbers. It is shown, to order $g^2$, that these 1/4 BPS operators have protected dimensions.

Polarization-Free Quantum Fields and Interaction

Abstract: A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

Systematics of quarter BPS operators in N=4 SYM

Abstract: A systematic construction is presented of 1/4 BPS operators in = 4 superconformal Yang-Mills theory, using either analytic superspace methods or components. In the construction, the operators of the classical theory annihilated by 4 out of 16 supercharges are arranged into two types. The first type consists of those operators that contain 1/4 BPS operators in the full quantum theory. The second type consists of descendants of operators in long unprotected multiplets which develop anomalous dimensions in the quantum theory. The 1/4 BPS operators of the quantum theory are defined to be orthogonal to all the descendant operators with the same classical quantum numbers. It is shown, to order g2, that these 1/4 BPS operators have protected dimensions.

The rogers–ramanujan recursion and intertwining operators

Abstract: We use vertex operator algebras and intertwining operators to study certain substructures of standard -modules, allowing us to conceptually obtain the classical Rogers–Ramanujan recursion. As a consequence we recover Feigin–Stoyanovsky's character formulas for the principal subspaces of the level 1 standard -modules.

L^2-Homology for von Neumann Algebras

Abstract: We define the notion of L^2 homology and L^2 Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L^2 homology for groups, using the ideas from the theory of correspondences. For the group algebra of a discrete group, our Betti numbers coincide with the L^2 Betti numbers of the group. We find a link between the first L^2 Betti number and free entropy dimension, which points to the non-vanishing of L^2 homology for the von Neumann algebra of a free group.

On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories

Abstract: Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense. In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique. As an important tool which is also interesting in its own right, we introduce a C � -algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.

Generalized jordan derivations on prime rings and standard operator algebras

Abstract: In this paper we initiate the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator algebras.