Independence in Operator Algebras

Various notions of independence of observables have been proposed within the algebraic framework of quantum field theory. We discuss relationships between these and the recently introduced notion of logical independence in a general operator-algebraic context. We show that C^*-independence implies an analogue of classical independence.     

Mirror Extensions of Vertex Operator Algebras

The mirror extensions for vertex operator algebras are studied. Two explicit examples of extensions of affine vertex operator algebras of type A are given which are not simple current extensions.     

Determinacy of States and Independence of Operator Algebras

The aim of this paper is to summarize, deepen,and comment upon recent results concerning states onoperator algebras and their extensions. The first partis focused on the relationship between pure states and singly generated subalgebras. Among otherswe show that every pure state on a separablealgebra A is uniquely determined by some element of Awhich exposes . The main part of this paper is the second section, dealing with characterizationof various types of independence conditions arising inthe axiomatics of quantum field theory. These twotopics, seemingly different, are connected by a common extension technique based on determinacy ofpure states.     

The q-Analogous Vertex Operator and Quantum Affine Algebras

The extended quantum affine algebras of q-analogous vertex operator have been constructed in this Jet ter, and the similar algebra structure of q-analogous fermionic vertex operator is also analysed.     

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.     

Affine Vertex Operator Algebras and Modular Linear Differential Equations

In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.     

Classes of Invariant Subspaces for Some Operator Algebras

New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace \({\mathcal{L}} \) such that all important subspace classes living on \({\mathcal{L}} \) are different. Moreover, we show that \({\mathcal{L}} \) can be chosen such that the set of σ-additive measures on subspace classes of \({\mathcal{L}} \) are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.     

Restricting and Extending States and Positive Maps on Operator Algebras

The aim of this paper is to summarize, deepen, and comment upon some recentresults concerning restrictions and extensions of states on operator algebras. Thefirst part is focused on the question of the circumstances under which a purestate or a completely positive map restricts to a pure state on maximal Abeliansubalgebra. In the second part we present an extension theorem forStone-algebra-valued measures on quotionts of JBW algebras and discuss its consequences.     

Vertex Operator Algebras Associated to Admissible Representations of

The Kac-Wakimoto admissible modules for are studied from the point of view of vertex operator algebras. It is shown that the vertex operator algebra L(l,0) associated to irreducible highest weight modules at admissible level is not rational if l is not a positive integer. However, a suitable change of the Virasoro algebra makes L(l,0) a rational vertex operator algebra whose irreducible modules are exactly these admissible modules for and for which the fusion rules are calculated. It is also shown that the q-dimensions with respect to the new Virasoro algebra are modular functions. Received: 4 April 1996/Accepted: 1 August 1996     

States on Operator Algebras and Axiomatic System of Quantum Theory

Recent development brings new results on the interplay of states on operator algebras and axiomatics of quantum mechanics. Neither hidden space in the sense of Kochen and Specker nor approximate hidden variables exist on von Neumann algebras. Tracial properties of states are connected with dispersions. The axioms on composite systems simplify to state extension properties.     

Generalized Operator Yang-Baxter Equations,Integrable ODEs and Nonassociative Algebras

Abstract

Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established.     

Framed Vertex Operator Algebras, Codes and the Moonshine Module

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ?, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ? are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras. Received: 14 July 1997 / Accepted: 8 September 1997     

Ergodic Actions of Universal Quantum Groups on Operator Algebras

We construct ergodic actions of compact quantum groups on C^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1) an ergodic action of the compact quantum A_u(Q) on the type III_u Powers factor R_u for an appropriate positive Q ] GL(2, Â); (2) an ergodic action of the compact quantum group A_u(n) on the hyperfinite II₁ factor R; (3) an ergodic action of the compact quantum group A_u(Q) on the Cuntz algebra _boxclose_boxclose{\cal O}_n for each positive matrix Q ] GL(n, ³); (4) ergodic actions of compact quantum groups on their homogeneous spaces, as well as an example of a non-homogeneous classical space that admits an ergodic action of a compact quantum group.     

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle. Received: 4 September 1996 / Accepted: 6 May 1997     

The Structure of Parafermion Vertex Operator Algebras: General Case

The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.     

The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given. As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between E ₇ and E ₈ lattice vertex operator algebras satisfy some modular differential equations. This result is an analogue of the result concerning the “hole” of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras E ₇ and E ₈.     

A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

Suppose that A ₁,…,A _N are observables (selfadjoint matrices) and ρ is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det {Cov  _ρ (A _j ,A _k )}, using the commutators [A _j ,A _k ]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [ρ,A _j ] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones.     

A Functional-Analytic Theory of Vertex (Operator) Algebras,II

For a finitely-generated vertex operator algebra V of central charge c, a locally convex topological completion H ^V is constructed. We construct on H ^V a structure of an algebra over the operad of the power Det ^c/2 of the determinant line bundle Det over the moduli space of genus-zero Riemann surfaces with ordered analytically parametrized boundary components. In particular, H ^V is a representation of the semi-group of the power Det ^c/2 (1) of the determinant line bundle over the moduli space of conformal equivalence classes of annuli with analytically parametrized boundary components. The results in Part I for -graded vertex algebras are also reformulated in terms of the framed little disk operad. Using Mays recognition principle for double loop spaces, one immediate consequence of such operadic formulations is that the compactly generated spaces corresponding to (or the k-ifications of) the locally convex completions constructed in Part I and in the present paper have the weak homotopy types of double loop spaces. We also generalize the results above to locally-grading-restricted conformal vertex algebras and to modules.