Structure Of Operator Algebras Research Articles

Uniqueness of VOA structure of 3C-algebra and 5A-algebra

The structure of 3C-algebra and 5A-algebra constructed by Lam-Yamada-Yamauchi is studied and the uniqueness of the vertex operator algebra structure of these two algebras is established. We also give the fusion rules for these two algebras.

Toverline{T} and EE, with implications for (A)dS subregion encodings

We initiate a study of subregion dualities, entropy, and redundant encoding of bulk points in holographic theories deformed by Toverline{T} and its generalizations. This includes both cut off versions of Anti de Sitter spacetime, as well as the generalization to bulk de Sitter spacetime, for which we introduce two additional examples capturing different patches of the bulk and incorporating the second branch of the square root dressed energy formula. We provide new calculations of entanglement entropy (EE) for more general divisions of the system than the symmetric ones previously available. We find precise agreement between the gravity side and deformed-CFT side results to all orders in the deformation parameter at large central charge. An analysis of the fate of strong subadditivity for relatively boosted regions indicates nonlocality reminiscent of string theory. We introduce the structure of operator algebras in these systems. The causal and entanglement wedges generalize to appropriate deformed theories but exhibit qualitatively new behaviors, e.g. the causal wedge may exceed the entanglement wedge. This leads to subtleties which we express in terms of the Hamiltonian and modular Hamiltonian evolution. Finally, we exhibit redundant encoding of bulk points, including the cosmological case.

6A-Algebra and its representations

In this paper, we study the structure and representation of a 6A-algebra which is a vertex operator algebra generated by two Ising vectors e,f with inner product 〈e,f〉=5210. In particular, we prove the uniqueness of the vertex operator algebra structure of this 6A-algebra, classify the irreducible modules, and determine the fusion rules.

Tingley's problem through the facial structure of operator algebras

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C⁎-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.

Lattice vertex algebras over fields of prime characteristic

Based on a work of Dong and Griess, we study a class of vertex algebras over fields of prime characteristic obtained from integral forms in lattice vertex operator algebras over complex field. In particular, the generating set is given, a bilinear form is obtained, and the simplicity is studied. Finally, the vertex operator algebra structure is characterized.

Operator Algebra of Foliations with Projectively Invariant Transverse Measure

We study the structure of operator algebras associated with the foliations which have projectively invariant measures. When a certain ergodicity condition on the measure preserving holonomies holds, the lack of holonomy invariant transverse measure can be established in terms of a cyclic cohomology class associated with the transverse fundamental cocycle and the modular automorphism group.

Hypotrochoids in conformal restriction systems and Virasoro descendants

A conformal restriction system is a commutative, associative, unital algebra equipped with a representation of the groupoid of univalent conformal maps on connected open sets of the Riemann sphere, along with a family of linear functionals on subalgebras, satisfying a set of properties including conformal invariance and a type of restriction. This embodies some expected properties of expectation values in conformal loop ensembles CLEκ (at least for 8/3 < κ ≤ 4). In the context of conformal restriction systems, we study certain algebra elements associated with hypotrochoid simple curves (a family of curves including the ellipse). These have the CLE interpretation of being ‘renormalized random variables’ that are nonzero only if there is at least one loop of hypotrochoid shape. Each curve has a center w, a scale ϵ and a rotation angle θ, and we analyze the renormalized random variable as a function of u = ϵeiθ and w. We find that it has an expansion in positive powers of u and , and that the coefficients of pure u () powers are holomorphic in w (). We identify these coefficients (the ‘hypotrochoid fields’) with certain Virasoro descendants of the identity field in conformal field theory, thereby showing that they form part of a vertex operator algebraic structure. This largely generalizes works by the author (in CLE), and the author with his collaborators Riva and Cardy (in SLE8/3 and other restriction measures), where the case of the ellipse, at the order u2, led to the stress–energy tensor of CFT. The derivation uses in an essential way the Virasoro vertex operator algebra structure of conformal derivatives established recently by the author. The results suggest in particular the exact evaluation of CLE expectations of products of hypotrochoid fields as well as nontrivial relations amongst them through the vertex operator algebra, and further shed light onto the relationship between CLE and CFT.

Transitivity and structure of operator algebras with a metric property

In this paper we discuss a new metric property that some operator algebras on Hilbert space possess and some resulting consequences concerning transitivity and structure theory of such algebras.

TOPOLOGICAL QUIVERS

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver [Formula: see text] is a C*-correspondence, and from this correspondence one may construct a Cuntz–Pimsner algebra [Formula: see text]. In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of [Formula: see text] can be determined from [Formula: see text]. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the gauge-invariant uniqueness theorem, the Cuntz–Krieger uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

Modular origin of chiral diffeomorphisms and their fuzzy analogs in higher-dimensional quantum field theories

The infinite-dimensional conformal symmetries beyond the Moebius-group (the vacuum symmetry) of chiral theories are shown to allow a natural derivation and interpretation in the operator-algebraic modular setting of algebraic QFT. In this way one learns that the diffeomorphisms of the circle beyond the Moebius group are bona fide Wigner symmetries with respect to suitably chosen non-vacuum state vector. The separation of chiral diffeomorphism symmetry from the issue of the structure of the energy–momentum tensor of 2-dimensional conformal theories permits the use of these symmetries in more general chiral theories which one encounters in the algebraic lightfront holography of massive d⩾1+1 dimensional QFTs. They only act locally on the lightfront projection but correspond to a new class of “fuzzy” symmetries in the original theory before the projection. These fuzzy symmetries remained hidden to quantization methods since they have a pure quantum origin; they do not require any spacetime noncommutativity and are in complete harmony with the causality and locality of the real time (non-commutative) operator algebra structure. Their use makes recent attempts to explain the desired universal structure of a possible future quantum Bekenstein-like area law in terms of Virasoro algebra structures on black hole horizons more palatable.

A Radon–Nikodým theorem for completely positive invariant multilinear maps and its applications

Noncommutative Radon–Nikodým theorems have attracted a great deal of attention in the theory of operator algebras. There has been considerable work on non-commutative Radon–Nikodým theorems not only for C*-algebras but also for algebras of unbounded operators [3, 5, 8, 9, 12, 13]. In this paper, we will develop a Radon–Nikodým type theorem for completely bounded and completely positive invariant multilinear maps.The concept of matricial order has turned out to be very important to understand the infinite-dimensional non-commutative structure of operator algebras. As the natural ordering attached to this structure, completely positive maps and completely bounded maps have been studied extensively. Results concerning completely bounded maps have many applications: cohomology of operator algebras, multipliers on group algebras, dilation theory, similarity theory, free product representations, and abstract characterizations of operator algebras.

Multiple-quantum operator algebra spaces and description for unitary time evolution of multilevel spin systems

A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.

Multiple-quantum operator algebra spaces and description for unitary time evolution of multilevel spin systems

A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.

Products of Decomposable Positive Operators

AbstractIn recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this by first showing that various results on operator ranges may be generalized to “measurable fields of operator ranges”.

THREE-POINT CORRELATION FUNCTIONS OF THE MINIMAL CONFORMAL THEORIES COUPLED TO 2D GRAVITY

The general three-point correlation functions of the minimal conformal field theories coupled to gravity are calculated using the specific Coulomb gas type quantization technique. The gravity interaction modifies in an essential way the operator algebra of the corresponding minimal model, in particular in canceling the decoupling of a finite number of primary fields, in case of rational theories. This is suggested to be related to the appearance of new physical states, under the BRST analysis of Lian and Zuckerman. Within the analytic technique, this effect is due to the corresponding singularities of the gravity sector operator algebra structure constants.

Some affine geometric aspects of operator algebras

The predual of a von Neumann algebra is shown to be a neutral strongly facially symmetric space, thereby suggesting an affine geometric approach to operator algebras and their non-associative analogues. Geometric proofs are obtained for the polar decompositions of normal functionals in ordered and non-ordered settings. A fundamental problem in the operator algebraic approach to quantum mechanics is to determine those algebraic structures in Banach spaces which are characterized by a set of geometrical axioms defining the quantum mechanical measuring process. This problem was solved in the context of ordered Banach spaces by Alfsen, Hanche-Olsen, and Shultz ([2], [1]) and led to a characterization of the state spaces of /2?*-algebras and C*-algebras. The main thrust of the present authors' recent research has been to find those algebraic structures induced on (unordered) Banach spaces in which such quantum mechanical axioms are satisfied. This project, which was initiated in [14] and [15] using the affine geometry of the dual unit ball, is used here to give a geometric proof of the Tomita-Sakai-Effros polar decomposition of a normal functional on a von Neumann algebra. Thus, the purpose of this partially expository paper is to show the richness and power of the affine geometric structure of the dual space of an operator algebra, by working in a purely geometric model. Indeed, since this geometry can be described in terms of the underlying real structure, it can be used to obtain new results in the real structure of operator algebras and in the structure of real operator algebras. For example, by using this approach, Dang ([5]) has shown that a raz/-linear isometry of a C*-algebra is the sum of a linear and a conjugate linear isometry, and hence is multiplicative, thereby obtaining a real analogue of Kadison's non-commutative extension of the BanachStone Theorem. The category of strongly facially symmetric (SFS) spaces (simply called facially symmetric spaces in [14] and [15]) has been shown to

Split faces and ideal structure of operator algebras.

Split faces and ideal structure of operator algebras.

On the ideal structure of operator algebras

On the ideal structure of operator algebras