Showing papers on "Operator algebra published in 2014"
TL;DR: In this paper, the authors point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction and suggest a tensor network calculation that may settle the issue.
Abstract: We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.
536 citations
TL;DR: In this article, a model-theoretic result was proved that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with non-principal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.
Abstract: We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.
153 citations
TL;DR: In this paper, it was shown that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions.
Abstract: We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well-known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.
98 citations
TL;DR: In this paper, the authors use continuous model theory to obtain several results concerning isomorphisms and embeddings between II-1 factors and their ultrapowers, including a poor man's resolution of the Connes embedding problem.
Abstract: We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.
78 citations
TL;DR: In this paper, Zhu's algebra, C 2 -algebra and C 2 cofiniteness of parafermion vertex operator algebras were studied and the classification of irreducible modules was established.
68 citations
TL;DR: The problem of characterizing when two multiplier algebras are (al- gebraically) isomorphic is also studied in this article, where it is shown that the converse does not hold for smooth curves.
Abstract: We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictionsMV of the multiplier algebraM of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We nd that MV is completely isometrically isomorphic toMW if and only if W is the image of V under a biholomorphic auto- morphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when d <1, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (al- gebraically) isomorphic is also studied. When V and W are each a nite union of irreducible varieties and a discrete variety in Bd with d <1, then an isomorphism betweenMV andMW deter- mines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak- continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold|particularly, smooth curves and Blaschke sequences.
62 citations
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 ⩽ 1 is discussed.
55 citations
TL;DR: In this paper, the authors use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics, and connect the higher index of the Dirac operator on a spin manifold with boundary to the higher Rho invariant on the boundary.
54 citations
TL;DR: In this paper, the authors consider the problem of defining a new Laplacian operator for the deformed algebra R 3, a deformation of the algebra of functions on R 3 which yields a foliation of R 3 into fuzzy spheres.
Abstract: We consider the noncommutative space R 3 , a deformation of the algebra of functions on R 3 which yields a foliation of R 3 into fuzzy spheres. We first review the construction of a natural matrix basis adapted to R 3 . We thus consider the problem of defining a new Laplacian operator for the deformed algebra. We propose an operator which is not of Jacobi type. The implication for field theory of the new Laplacian is briefly discussed.
53 citations
TL;DR: In this paper, it was shown that AW*-algebras are equivalent to a subcategory of active lattices, which are formed from three ingredients: symmetry, symmetry, and action of the latter on the former.
49 citations
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TL;DR: The most recent wave of applications of logic to operator algebras is a young and rapidly developing field as mentioned in this paper, which is a snapshot of the current state of the art.
Abstract: The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.
TL;DR: In this paper, the authors introduced the notion of intermediate vertex subalgebras of lattice vertex operator algebra, which is a generalization of the principal subspaces.
Abstract: 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 E7 and E8 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 E7 and E8.
TL;DR: In this paper, the effect of failure of fusion at the level of Verlinde products was studied for the (p+,p−)(p+p−) triplet algebras.
TL;DR: In this article, the authors introduce the Drury-Arveson space and give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.
Abstract: The Drury-Arveson space $H^2_d$, also known as symmetric Fock space or the $d$-shift space, is a Hilbert function space that has a natural $d$-tuple of operators acting on it, which gives it the structure of a Hilbert module. This survey aims to introduce the Drury-Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.
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TL;DR: In this article, the intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g. the Poincare pairing) is described in terms of representation theory of some vertex operator algebras.
Abstract: We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincare pairing) in terms of representation theory of some vertex operator algebras ("$\mathscr W$-algebras").
TL;DR: In this paper, a sharpened version of a conjecture of Dong-Mason about lattice subalgebras of a strongly regular vertex operator algebra V was shown to be true.
Abstract: We prove a sharpened version of a conjecture of Dong–Mason about lattice subalgebras of a strongly regular vertex operator algebra V, and give some applications. These include the existence of a canonical conformal vertex operator subalgebra W ⊗ G ⊗ Z of V, and a generalization of the theory of minimal models.
TL;DR: In this paper, the concept of an embeddable quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups.
TL;DR: In this paper, an equivalence between two Hilbert spaces, namely the space of ground states of strings on an associated mapping torus with Tcffff 2 fiber, and the ground states on a twisted circle compactification of U(1) gauge theory with a certain class of tridiagonal matrices of coupling constants (with corners) was established.
Abstract: We develop an equivalence between two Hilbert spaces: (i) the space of states of U(1)
n
Chern-Simons theory with a certain class of tridiagonal matrices of coupling constants (with corners) on T
2; and (ii) the space of ground states of strings on an associated mapping torus with T
2 fiber. The equivalence is deduced by studying the space of ground states of SL(2, ℤ)-twisted circle compactifications of U(1) gauge theory, connected with a Janus configuration, and further compactified on T
2. The equality of dimensions of the two Hilbert spaces (i) and (ii) is equivalent to a known identity on determinants of tridiagonal matrices with corners. The equivalence of operator algebras acting on the two Hilbert spaces follows from a relation between the Smith normal form of the Chern-Simons coupling constant matrix and the isometry group of the mapping torus, as well as the torsion part of its first homology group.
07 Aug 2014
TL;DR: In this paper, it was shown that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problems.
Abstract: In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problem. Moreover, we relate the computability of the operator norm on the group F2 ×F2 to Kirchberg’s QWEP Conjecture, a fundamental open problem in the theory of operator algebras.
TL;DR: In this paper, a new method for obtaining branching rules for affine Kac-Moody Lie algebras at negative integer levels is developed, which enables decomposing certain and -modules at negative levels.
Abstract: We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type , obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for l ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type . Next, we notice that the category of modules at level -2l + 3 has the isomorphic fusion algebra. This enables us to decompose certain and -modules at negative levels.
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TL;DR: The irreducible modules for the parafermion vertex operator algebra associated to any finite dimensional Lie algebra and any positive integer are identified, the quantum dimensions are computed and the fusion rules are determined as discussed by the authors.
Abstract: The irreducible modules for the parafermion vertex operator algebra associated to any finite dimensional Lie algebra and any positive integer are identified, the quantum dimensions are computed and the fusion rules are determined
TL;DR: In this paper, a new notion of positivity in operator algebras with and without contractive approximate identities (cais) has been introduced, with an eye to extending certain C*-algebraic results and theories to more general algesbras.
Abstract: We continue our study of operator algebras with and contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algebra and the C*-algebra it generates. In much of this it is not necessary that the algebra have an approximate identity. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.
TL;DR: In this article, the authors consider principal subspaces W L ( k Λ 0 ) and W N ( k ǫ 0 ) of a standard module L (k Λ ) and a generalized Verma module N (k à 0 ) at level k ≥ 1 for affine Lie algebras of type B 2 (1 ) and find combinatorial bases in terms of quasi-particles.
TL;DR: In this paper, the authors introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation, in which one chooses the degrees of the cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation.
Abstract: We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous. In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables. We apply these results to show that the quantum Grassmannians $K_q[Gr(k, n)]$ admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geis–Leclerc–Schroer and completes earlier work of the authors on the finite-type cases.
TL;DR: In this article, the fusion rules of the free wreath product quantum groups Γ ˆ ≀ ⁎ S N + for any discrete group Γ are given.
TL;DR: This work considers the category W* of W*-algebras together with normal sub-unital maps, provides an order-enrichment for this category and exhibit a class of its endofunctors with a canonical fixpoint.
TL;DR: In this paper, the authors studied subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras.
TL;DR: In this article, a detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group is given, and the full proof of the functional relations in the form independent of the representation of quantum group on the quantum space is presented.
Abstract: A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_2))$ is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described. This is a degression of the corresponding consideration for the case of the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$ with an extensions to the higher spin case.
TL;DR: In this article, the authors considered non-equilibrium quantum steady states in conformal field theory (CFT) on star-graph configurations, with a particular, simple connection condition at the vertex of the graph.
Abstract: We consider non-equilibrium quantum steady states in conformal field theory (CFT) on star-graph configurations, with a particular, simple connection condition at the vertex of the graph. These steady states occur after a large time as a result of initially thermalizing the legs of the graph at different temperatures, and carry energy flows. Using purely Virasoro algebraic calculations we evaluate the exact scaled cumulant generating function for these flows. We show that this function satisfies a generalization of the usual non-equilibrium fluctuation relations. This extends results by two of the authors to the case of more than two legs. It also provides an alternative derivation centered on Virasoro algebra operators rather than local fields, hence an alternative regularization scheme, thus confirming the validity and universality of the scaled cumulant generating function. Our derivation shows how the usual Virasoro algebra leads, in large volumes, to a continuous-index Virasoro algebra for which we develop diagrammatic principles, which may be of interest in other non-equilibrium contexts as well. Finally, our results shed light on the Poisson-process interpretation of the long-time energy transfer in CFT.
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TL;DR: In this paper, the fusion rules for the free wreath product quantum groups of Kac type were derived based on a combinatorial description of the intertwiner spaces between certain generating representations of the groups.
Abstract: In this paper, we find the fusion rules for the free wreath product quantum groups $\mathbb{G}\wr_*S_N^+$ for all compact matrix quantum groups of Kac type $\mathbb{G}$ and $N\ge4$. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of $\mathbb{G}\wr_*S_N^+$. The combinatorial properties of the intertwiner spaces in $\mathbb{G}\wr_*S_N^+$ then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between $\mathbb{G}\wr_*S_N^+$ and a compact quantum group whose dual is a discrete quantum subgroup of the free product $\widehat{\mathbb{G}}*\widehat{SU_q(2)}$, for some $0