Showing papers on "Operator algebra published in 2013"
TL;DR: In this paper, a continuous functional calculus in quaternionic Hilbert spaces is defined, starting from basic issues regarding the notion of spherical spectrum of a normal operator, and several versions of the spectral map theorem are proved also for unbounded operators.
Abstract: The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.
212 citations
Book•
04 Apr 2013
TL;DR: In this paper, the authors present a general theory of topological *-algebras and their application in the context of representation theory, including Hermitian and symmetric topology.
Abstract: Introduction. Part I: General Theory. I. Background material. II. Locally C* -algebras. III. Representation theory. IV. Structure space of an m* -convex algebra. V. Hermitian and symmetric topological *-algebras. Part II: Applications. VI. Integral representations. Uniqueness of topology. VII. Tensor products of topological *-algebras. Bibliography.
149 citations
TL;DR: In this article, the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group associated with a Weyl group element w has the structure of a quantum cluster algebra.
Abstract: We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac–Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory \({\fancyscript{C}_{w}}\) of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities that can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.
142 citations
TL;DR: In this paper, the quantum dimensions of vertex operator algebras are defined and their properties are discussed systematically, and a criterion for simple current modules of a rational vertex operator algebra is given.
Abstract: The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed systematically. The quantum dimensions of the Heisenberg vertex operator algebra modules, the Virasoro vertex operator algebra modules and the lattice vertex operator algebra modules are computed. A criterion for simple current modules of a rational vertex operator algebra is given. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A full Galois theory for rational vertex operator algebras is established using the quantum dimensions.
107 citations
TL;DR: In this paper, the authors study the braided monoidal structure that the fusion product induces on the Abelian category -mod, the category of representations of the triplet W-algebra.
Abstract: We study the braided monoidal structure that the fusion product induces on the Abelian category -mod, the category of representations of the triplet W-algebra . The -algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalize the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of -mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of -mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective -modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.
95 citations
TL;DR: In this article, a one-parameter family of algebras FIO ( Ξ, s ), 0 ⩽ s⩽ ∞, consisting of Fourier integral operators is constructed, which is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation.
77 citations
TL;DR: In this article, it was shown that every unital operator system has sufficiently many boundary representations to generate the C*-envelope, which is the boundary representation of a unital unital algebra.
Abstract: We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C*-envelope.
73 citations
TL;DR: In this paper, a logarithmic conformal field theory (CFT) is defined, where the energy operator fails to be diagonalisable on the quantum state space, the CFT is defined as one whose quantum space of states is constructed from a collection of representations including reducible but indecomposable ones.
Abstract: Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: Schramm-Loewner evolution and Smirnov’s discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U(1|1) Wess-Zumino-Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie’s 1993 article (his paper also contains the first usage of the term “logarithmic conformal field theory”). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more complicated non-rational theories. Examples include critical percolation, supersymmetric string backgrounds, disordered electronic systems, sandpile models describing avalanche processes, and so on. In each case, the non-rationality and non-unitarity of the CFT suggested that a more general theoretical framework was needed. Driven by the desire to better understand these applications, the mid-nineties saw significant theoretical advances aiming to generalise the constructs of rational CFT to a more general class. In 1994, Nahm introduced an algorithm for computing the fusion product of representations which was significantly generalised two years later by Gaberdiel and Kausch who applied it to explicitly construct (chiral) representations upon which the energy operator acts non-diagonalisably. Their work made it clear that underlying the physically relevant correlation functions are classes of reducible but indecomposable representations that can be investigated mathematically to the benefit of applications. In another direction, Flohr had meanwhile initiated the study of modular properties of the characters of logarithmic CFTs, a topic which had already evoked much mathematical interest in the rational case. Since these seminal theoretical papers appeared, the field has undergone rapid development, both theoretically and with regard to applications. Logarithmic CFTs are now known to describe non-local observables in the scaling limit of critical lattice models, for example percolation and polymers, and are an integral part of our understanding of quantum strings propagating on supermanifolds. They are also believed to arise as duals of three-dimensional chiral gravity models, fill out hidden sectors in non-rational theories with non-compact target spaces, and describe certain transitions in various incarnations of the quantum Hall effect. Other physical
70 citations
TL;DR: In this article, it was shown that the negative answer to each of these questions is equivalent to the Continuum Hypothesis, extending results of Ge{Hadwin and the rst author.
Abstract: Several authors have considered whether the ultrapower and the relative com- mutant of a C*-algebra or II1 factor depend on the choice of the ultralter. We show that the negative answer to each of these questions is equivalent to the Continuum Hypothesis, extending results of Ge{Hadwin and the rst author.
70 citations
TL;DR: This paper proposes a formulation of the problem of Gian-Carlo Rota's problem using the framework of operated algebras and viewing an associative algebra with a linear operator as one that satisfies a certain operated polynomial identity, and allows to apply theories of rewriting systems and Grobner-Shirshov bases.
63 citations
TL;DR: In this article, the algebraic approach to quantum field theory on curved backgrounds is introduced, based on a set of axioms, first written down by Haag and Kastler, which consists of a two-step procedure.
Abstract: Goal of this review is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, a suitable algebra of observables is assigned to a physical system, which is meant to encode all algebraic relations among observables, such as commutation relations, while, in the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give to the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.
TL;DR: It is shown that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.
TL;DR: In this article, the authors construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity and show that the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant ''generalized ladder operators'' recently introduced in the $\mathrm{U}(N)$ approach.
Abstract: We construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant ``generalized ladder operators'' recently introduced in the $\mathrm{U}(N)$ approach to intertwiners and spin networks. We comment on quantization ambiguities that appear in the definition of the holonomy operator and use these ambiguities as a toy model to test a class of quantization ambiguities which is present in the standard regularization and definition of the Hamiltonian constraint operator in loop quantum gravity.
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TL;DR: In this paper, the authors define and study full and reduced crossed products of algebras of operators on ϵ-finite $L^p$ spaces by isometric actions of second countable locally compact groups.
Abstract: For $p \in [1, \infty),$ we define and study full and reduced crossed products of algebras of operators on $\sigma$-finite $L^p$ spaces by isometric actions of second countable locally compact groups. We give universal properties for both crossed products. When the group is abelian, we prove the existence of a dual action on the full and reduced $L^p$ operator crossed products. When the group is discrete, we construct a conditional expectation to the original algebra which is faithful in a suitable sense. For a free action of a discrete group on a compact metric space $X,$ we identify all traces on the reduced $L^p$ operator crossed product, and if the action is also minimal we show that the reduced $L^p$ operator crossed product is simple. We prove that the full and reduced $L^p$ operator crossed products of an amenable $L^p$ operator algebra by a discrete amenable group are again amenable. We prove a Pimsner-Voiculescu exact sequence for the K-theory of reduced $L^p$ operator crossed products by ${\mathbb{Z}}.$ We show that the $L^p$ analogs ${\mathcal{O}}_d^p$ of the Cuntz algebras ${\mathcal{O}}_d$ are stably isomorphic to reduced $L^p$ operator crossed products of stabilized $L^p$ UHF algebra by ${\mathbb{Z}},$ and show that $K_0 ({\mathcal{O}}_d^p) \cong {\mathbb{Z}} / (d - 1) {\mathbb{Z}}$ and $K_1 ({\mathcal{O}}_d^p) = 0.$
Book•
07 Feb 2013
TL;DR: The first four axioms of QM are propositions, quantum states and observables as mentioned in this paper, and Spectral Theory I: generalities, abstract C -algebras and operators in B(H), Spectral theory II: unbounded operators on Hilbert spaces.
Abstract: Introduction and mathematical backgrounds.- Normed and Banach spaces, examples and applications.- Hilbert spaces and bounded operators.- Families of compact operators on Hilbert spaces and fundamental properties.- Densely-defined unbounded operators on Hilbert spaces.- Phenomenology of quantum systems and Wave Mechanics: an overview.- The first 4 axioms of QM: propositions, quantum states and observables.- Spectral Theory I: generalities, abstract C -algebras and operators in B(H).- Spectral theory II: unbounded operators on Hilbert spaces.- Spectral Theory III: applications.- Mathematical formulation of non-relativistic Quantum Mechanics.- Introduction to Quantum Symmetries.- Selected advanced topics in Quantum Mechanics.- Introduction to the Algebraic Formulation of Quantum Theories.- Order relations and groups.- Elements of differential geometry.
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TL;DR: This work gives an overview of analytic tools for the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large.
Abstract: This work gives an overview of analytic tools for the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large. Tools from probability theory, operator algebra, and statistical physics are reviewed. The survey of analytical tools is complemented by examples of applications in communications engineering. Asymptotic eigenvalue distributions of many classes of random matrices are given. The treatment includes the problem of moments and the introduction of the Stieltjes transform. Free probability theory, which evolved from non-commutative operator algebras, is explained from a probabilistic point of view in order to better fit the engineering community. For that purpose freeness is defined without reference to non-commutative algebras. The treatment includes additive and multiplicative free convolution, the R-transform, the S-transform, and the free central limit theorem. The replica method developed in statistical physics for the purpose of analyzing spin glasses is reviewed from the viewpoint of its applications in communications engineering. Correspondences between free energy and mutual information as well as energy functions and detector metrics are established. These analytic tools are applied to the design and the analysis of linear multiuser detectors, the modelling of scattering in communication channels with dual antennas arrays, and the analysis of optimal detection for communication via code-division multiple-access and/or dual antenna array channels.
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TL;DR: In this paper, it is proved that most well-known rational vertex operator algebras are unitary and the classification of unitary vertex operators with central charge c less than or equal to 1 is discussed.
Abstract: Unitary vertex operator algebras are introduced and studied. It is proved that most well-known rational vertex operator algebras are unitary. The classification of unitary vertex operator algebras with central charge c less than or equal to 1 is also discussed.
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TL;DR: In this article, it was shown that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C u -categories over ReppGq up to natural equivalence.
Abstract: An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module structure over the tensor category ReppGq of finite-dimensional representations of G. We show that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C u -categories over ReppGq up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts.
TL;DR: In this article, the authors studied a class of polynomials in non-commutative indeterminates Z i, j, i ∈ { 1, …, k }, j ∈ [ 1, …, n i ] and showed that the characteristic function is a complete unitary invariant for the class of completely non-coisometric elements.
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TL;DR: In this article, a variety of non-selfadjoint operator algebras that depend on the choice of the co- variance relation, along with the smallest C -algebra they generate, namely the C � -envelope, are examined.
Abstract: Given a dynamical system (A,�) where A is a unital C � -algebra andis a (possibly nonunital) ∗-endomorphism of A, we examine families (�,{Ti}) such thatis a representation of A, {Ti} is a Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of nonselfadjoint operator algebras that depend on the choice of the co- variance relation, along with the smallest C � -algebra they generate, namely the C � -envelope. We then relate each occurrence of the C�-envelope to (a full corner of) an appropriate twisted crossed product. We provide a counterexample to show the extent of this variety. In the context of C�-algebras, these results can be interpreted as analogues of Stacey's famous result, for nonautomorphic systems and n > 1. Our study involves also the one variable generalized crossed products of Stacey and Exel. In particular, we refine a result that appears in the pioneering paper of Exel on (what is now known as) Exel systems.
TL;DR: In this paper, the authors compare the two topos-theoretic approaches to quantum mechanics, the contravariant approach and the covariant approach, using the topos of presheaves on a specific context category, defined as the poset of commutative von Neumann subalgebras of some given von NE algebra.
Abstract: The aim of this paper is to compare the two topos-theoretic approaches to quantum mechanics that may be found in the literature to date. The first approach, which we will call the contravariant approach, was originally proposed by Isham and Butterfield, and was later extended by Doring and Isham. The second approach, which we will call the covariant approach, was developed by Heunen, Landsman and Spitters. Motivated by coarse-graining and the Kochen-Specker theorem, the contravariant approach uses the topos of presheaves on a specific context category, defined as the poset of commutative von Neumann subalgebras of some given von Neumann algebra. In particular, the approach uses the spectral presheaf. The intuitionistic logic of this approach is given by the (complete) Heyting algebra of closed open subobjects of the spectral presheaf. We show that this Heyting algebra is, in a natural way, a locale in the ambient topos, and compare this locale with the internal Gelfand spectrum of the covariant approach. In the covariant approach, a non-commutative C*-algebra (in the topos Set) defines a commutative C*-algebra internal to the topos of covariant functors from the context category to the category of sets. We give an explicit description of the internal Gelfand spectrum of this commutative C*-algebra, from which it follows that the external spectrum is spatial. Using the daseinisation of self-adjoint operators from the contravariant approach, we give a new definition of the daseinisation arrow in the covariant approach and compare it with the original version. States and state-proposition pairing in both approaches are compared. We also investigate the physical interpretation of the covariant approach.
TL;DR: In this paper, an algebraic refinement of a functor from the category of number fields to Bost-Connes systems has been constructed by Laca, Neshveyev and Trifkovic, using a base-change functor for a class of algebraic endomotives.
Abstract: This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes (Sel. Math. 1(3):411–457, 1995). In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Consani and Marcolli (Adv. Math. 214(2):761–831, 2007), and using a classification result of Borger and de Smit (arXiv:1105.4662) for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreover the uniqueness of the arithmetic model is shown by Sergey Neshveyev in an appendix. In the second part of the paper we introduce a base-change functor for a class of algebraic endomotives and construct in this way an algebraic refinement of a functor from the category of number fields to the category of Bost-Connes systems, constructed recently by Laca, Neshveyev and Trifkovic (arXiv:1010.4766).
TL;DR: The main aim of the lecture notes is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites as discussed by the authors, which can be used, for example, to model the materials in condensed matter physics.
Abstract: This is an extended and corrected version of lecture notes originally written for a one semester course at Leibniz University Hannover. The main aim of the notes is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites. Such systems can be used, for example, to model the materials in condensed matter physics. The notes provide the necessary background material to access recent literature in the field. Some of these recent results are also discussed.
The contents are roughly as follows: (1) quick recap of essentials from functional analysis, (2) introduction to operator algebra, (3) algebraic quantum mechanics, (4) infinite systems (quasilocal algebra), (5) KMS and ground states, (6) Lieb-Robinson bounds, (7) algebraic quantum field theory, (8) superselection sectors of the toric code, (9) Haag-Ruelle scattering theory in spin systems, (10) applications to gapped phases.
The level is aimed at students who have at least had some exposure to (functional) analysis and have a certain mathematical "maturity".
TL;DR: The null-plane Gell-Mann-Oakes-Renner formula is derived in this article, and a general prescription is given for mapping all chiral-symmetry breaking QCD condensates to chiral symmetry conserving nullplane QCD Condensates.
14 May 2013
TL;DR: A new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light is proposed and the definition of a linear space is given and many properties about its operations are proved, e.g., addition and scalar multiplication.
Abstract: Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.
TL;DR: In this paper, it was shown that every local Lie derivation of a map of a Banach space of dimension > 2 is a 2-local derivation if and only if it has the form \({A \mapsto AT - TA + \psi(A)}\), where T is a homogeneous map from B(X) into ψ, and ψ is the sum of commutators.
Abstract: Let X be a Banach space of dimension > 2. We show that every local Lie derivation of B(X) is a Lie derivation, and that a map of B(X) is a 2-local Lie derivation if and only if it has the form \({A \mapsto AT - TA + \psi(A)}\), where \({T \in B(X)}\) and ψ is a homogeneous map from B(X) into \({\mathbb{F}I}\) satisfying \({\psi(A + B) = \psi(A)}\) for \({A, B \in B(X)}\) with B being a sum of commutators.
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TL;DR: In this article, it was shown that if U and W are C_1-cofinite Ω(n)-gradable vertex operator algebra modules, then the fusion product is defined by (logarithmic) intertwining operators.
Abstract: Let V be a vertex operator algebra. We prove that if U and W are C_1-cofinite {\mathbb N}-gradable V-modules, then a fusion product U\boxtimes W is well-defined and also a C_1-cofinite {\mathbb N}-gradable V-module, where the fusion product is defined by (logarithmic) intertwining operators. This is also true for C_2-cofinite {\mathbb N}-gradable modules.
TL;DR: In this article, a rotationally invariant space non-commutativity (NC) space Rλ3, an analog of the Coulomb problem configuration space (R3 with the origin excluded) is introduced.
Abstract: The aim of this paper is to find out how it would be possible for space non-commutativity (NC) to alter the quantum mechanics (QM) solution of the Coulomb problem. The NC parameter λ is to be regarded as a measure of the non-commutativity – setting λ = 0 which means a return to the standard quantum mechanics. As the very first step a rotationally invariant NC space Rλ3, an analog of the Coulomb problem configuration space (R3 with the origin excluded) is introduced. Rλ3 is generated by NC coordinates realized as operators acting in an auxiliary (Fock) space F. The properly weighted Hilbert-Schmidt operators in F form Hλ, a NC analog of the Hilbert space of the wave functions. We will refer to them as “wave functions” also in the NC case. The definition of a NC analog of the hamiltonian as a hermitian operator in Hλ is one of the key parts of this paper. The resulting problem is exactly solvable. The full solution is provided, including formulas for the bound states for E < 0 and low-energy scattering fo...
TL;DR: In this article, a pseudo-trace function for vertex operator algebras satisfying Zhu's finiteness condition was proposed and applied to the ℤ2-orbifold model associated with d-pairs of symplectic fermions.
Abstract: We propose a method to give pseudo-trace functions for vertex operator algebras satisfying Zhu's finiteness condition and apply our method to the ℤ2-orbifold model associated with d-pairs of symplectic fermions. Pseudo-trace functions are defined by using symmetric linear functions on (higher) Zhu's algebras. But our approach does not use such algebras. For d = 1, we determine the dimension of the space of one-point functions, that is, the one of the triplet -algebra. For d > 1, we construct 22d-1 + 3 linearly independent one-point functions and study their values at the vacuum vector.
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TL;DR: In this article, the authors explore the possibility of characterising C* algebras by their (non-isometric) Banach algebra structure alone, and introduce a property of Banach algesbras, the Total Reduction Property, and conjecture that a Banach Algebra has this property if and only if it is isomorphic to a C* Algebra.
Abstract: In this thesis we explore the the possibility of characterising C* algebras by their (non-isometric) Banach algebra structure alone.
We introduce a property of Banach algebras, the Total Reduction Property, and conjecture that a Banach algebra has this property if and only if it is isomorphic to a C* algebra. We establish some partial results in support of this conjecture.