Abstract: Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes.

Abstract: We investigate trace functions of modules for vertex operator algebras (VOA) satisfying C2 -cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) A(V) is semisimple and (2) C2 -cofiniteness. We show that C2 -cofiniteness is enough to prove a modular invariance property. For example, if a VOA V= ⊕ m=0 ∞ V m is C2 -cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of V -modules is a finite-dimensional $\SL_2(\mathbb{Z})$ SL 2 ( Z) -invariant space and the central charge and conformal weights are all rational numbers. Namely, we show that C2 -cofiniteness implies "rational conformal field theory" in a sense as expected in Gaberdiel and Neitzke [GN]. Viewing a trace map as a symmetric linear map and using a result of symmetric algebras, we introduce "pseudotraces" and pseudotrace functions and then show that the space spanned by such pseudotrace functions has a modular invariance property. We also show that C2 -cofiniteness is equivalent to the condition that every weak module is an N -graded weak module that is a direct sum of generalized eigenspaces of L(0) .

Abstract: We assemble the spectrum of single-trace operators in free N = 4 SU(N) SYM theory into irreducible representations of the Higher Spin symmetry algebra hs(2,2|4). Higher Spin representations or YT-pletons are associated to Young tableaux (YT) corresponding to representations of the symmetric group compatible with the cyclicity of color traces. After turning on interactions gYM 6 0, YT-pletons decompose into infinite towers of representations of the superconformal algebra psu(2,2|4) and anomalous dimensions are generated. We work out the decompositions of tripletons with respect to the N = 4 superconformal algebra psu(2,2|4) and compute their anomalous dimensions to lowest non-trivial order in g 2 YMN at large N. We then focus on operators/states sitting in semishort multiplets of psu(2,2|4). By passing them through a semishort-sieve that removes superdescendants, we derive compact expressions for the partition function of semishort primaries.

Abstract: We analyse the renormalisation group flow for D-branes in WZW models from the point of view of the boundary states. To this end we consider loop operators that perturb the boundary states away from their ultraviolet fixed points, and show how to regularise and renormalise them consistently with the global symmetries of the problem. We pay particular attention to the chiral operators that only depend on left-moving currents, and which are attractors of the renormalisation group flow. We check (to lowest non-trivial order in the coupling constant) that at their stable infrared fixed points these operators measure quantum monodromies, in agreement with previous semiclassical studies. Our results help clarify the general relationship between boundary transfer matrices and defect lines, which parallels the relation between (non-commutative) fields on (a stack of) D-branes and their push-forwards to the target-space bulk.

Abstract: We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is isomorphic to V as a V-module. (ii) Every weak V-module gradable by nonnegative integers is completely reducible. (iii) V is C_2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Then the matrices formed by the fusion rules among the irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation \tau\mapsto -1/\tau on the space of characters of irreducible V-modules. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation \tau\mapsto -1/\tau is symmetric.

Abstract: We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincare-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three-dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

Abstract: Within the algebraic setting of quantum field theory, a condition is given which implies that the intersection of algebras generated by field operators localized in wedge-shaped regions of the two-dimensional Minkowski space is non-trivial; in particular, there exist compactly localized operators in such theories which can be interpreted as local observables. The condition is based on spectral (nuclearity) properties of the modular operators affiliated with wedge algebras and the vacuum state and is of interest in the algebraic approach to the formfactor program, initiated by Schroer. It is illustrated here in a simple class of examples.

Abstract: We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$ The framework is that of {\it odd unbounded} $\theta$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$ In the type $II_{\infty}$ case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known We borrow Ezra Getzler's idea (suggested by I M Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space Applications in both the type I and type II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the $K$-homology and $K$-theory of $\mathcal A$

Abstract: Conformal quantum field theory on the half-space x>0 of Minkowski space-time ("boundary CFT") is analyzed from an algebraic point of view, clarifying in particular the algebraic structure of local algebras and the bi-localized charge structure of local fields. The field content and the admissible boundary conditions are characterized in terms of a non-local chiral field algebra.

Abstract: One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type ${\rm III}_{1}$ factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other.

Abstract: We give a survey of cyclic homology/cohomology theory including a detailed discussion of cyclic theories for various classes of topological algebras. We show how to associate cyclic classes with Fredholm modules and K-theory classes and how to construct a completely general bivariant Chern-Connes character from bivariant K-theory to bivariant cyclic theory.

Abstract: Preface 1. Sets and structures 2. Groups 3. Vector spaces 4. Linear operators and matrices 5. Inner product spaces 6. Algebras 7. Tensors 8. Exterior algebra 9. Special relativity 10. Topology 11. Measure theory and integration 12. Distributions 13. Hilbert space 14. Quantum theory 15. Differential geometry 16. Differentiable forms 17. Integration on manifolds 18. Connections and curvature 19. Lie groups and lie algebras.

Abstract: We consider the Markov operator Pφ on a discrete quantum group given by convolution with a q-tracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on Aφ. The C ∗-algebraAφ has a natural time evolution, and the unit can always be represented by a KMS state. Any such state gives rise to a u.c.p. map from the von Neumann closure of Aφ in its GNS representation to the von Neumann algebra of bounded harmonic elements, which is an analogue of the Poisson integral. Under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual. Next we apply these results to identify the Martin boundary of the dual of SUq(2) with the quantum homogeneous sphere of Podleś. This result extends and unifies previous results by Ph. Biane and M. Izumi.

Abstract: A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal R$ on a standard Borel space is amenable if and only if the full group $[{\mathcal R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial space in the sense of Gromov occuring in the automorphism groups of injective factors of type $III$.

Abstract: We analyze the decomposition rank (a notion of covering dimension for nuclear C*-algebras introduced by E. Kirchberg and the author) of subhomogeneous C*-algebras. In particular, we show that a subhomogeneous C*-algebra has decomposition rank $n$ if and only if it is recursive subhomogeneous of topological dimension $n$, and that $n$ is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C. Phillips to show the following. Let $A$ be the crossed product C*-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then the decomposition rank of $A$ is dominated by the covering dimension of the underlying manifold.

Abstract: We consider a condition for non-degenerate commuting squares of matrix algebras (finite dimensional von Neumann algebras) called the \emph{span condition}, which in the case of the $n$-dimensional standard spin models is shown to be satisfied if and only if $n$ is prime. We prove that the commuting squares satisfying the span condition are isolated among all commuting squares (modulo isomorphisms). In particular, they are finiteley many for any fixed dimension. Also, we give a conceptual proof of previous constructions of certain one-parameter families of biunitaries.

Abstract: In this paper we study representations of conformal nets associated with positive definite even lattices and their orbifolds with respect to isometries of the lattices. Using previous general results on orbifolds, we give a list of all irreducible representations of the orbifolds, which generate a unitary modular tensor category.

Abstract: The functor Max of Mulvey assigns to each unital C*-algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*-algebras up to ∗-isomorphism. In this paper we provide a proof of this and of the stronger fact that for every isomorphism u : Max A → Max B of unital involutive quantales there is a ∗-isomorphism b : A → B such that Max b u coincides with u when restricted to the left-sided elements of Max A. But we also show that isomorphisms u : Max A → Max B may exist for which no isomorphism v : A → B is such that Max v = u.

Abstract: We present a closed formula for a family of star-products by replacing the partial derivatives in the Moyal-Weyl formula with commuting vector fields. We show how to reproduce algebra relations on commutative spaces with these star-products and give some physically interesting examples of that procedure.

Abstract: k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a satisfactory version of the usual topological classification in terms of subgroups of a fundamental group. We then use this classification to describe the C*-algebras of covering k-graphs as crossed products by coactions of homogeneous spaces, generalizing recent results on the C*-algebras of graphs.

Abstract: We study the asymmetric simple exclusion process with open boundaries and derive the exact form of the joint probability function for the occupation number and the current through the system. We further consider the thermodynamic limit, showing that the resulting distribution is non-Gaussian and that the density fluctuations have a discontinuity at the continuous phase transition, while the current fluctuations are continuous. The derivations are performed by using the standard operator algebraic approach and by the introduction of new operators satisfying a modified version of the original algebra.

Abstract: The algebra of diffeomorphisms derived from general coordinate transformations on commuting coordinates is represented by differential operators on noncommutative spaces. The algebra remains unchanged, the comultiplication however is deformed, that way we have found a deformed bialgebra of diffeomorphisms. Scalar, vector and tensor fields are defined with appropriate transformation laws under the deformed algebra and a differential calculus is developed. For pedagogical reasons the formalism is developed for the $\theta$-deformed space as it is the best known example of deformed spaces.

Abstract: The goal of this paper is to study the BMN correspondence in the fermionic sector. On the field theory side, we compute matrix elements of the dilatation operator in = 4 Super Yang-Mills for BMN operators containing two fermion impurities. Our calculations are performed up to and including (λ') in the 't Hooft coupling and (g2) in the Yang-Mills genus counting parameter. On the string theory side, we compute the corresponding matrix elements of the interacting string hamiltonian in string field theory, using the three-string interaction vertex constructed by Spradlin and Volovich (and subsequently elaborated by Pankiewicz and Stefanski). In string theory we use the natural string basis, and in field theory the basis which is isomorphic to it. We find that the matrix elements computed in field theory and the corresponding string amplitudes derived from the three-string vertex are, in all cases, in perfect agreement.

Abstract: We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus three at a fifth root of unity, we still give an explicit basis. Integrality properties of Witten-Reshetikhin-Turaev quantum invariants of 3- manifolds have been studied intensively in the last several years. H. Murakami (Mu1, Mu2) showed that the SU(2)- and SO(3)-invariants at a root of unity q of prime order are algebraic integers. This was reproved in (MR2) and generalised to all classical Lie types in (MW, TY) and then to all Lie types in (Le). These integrality properties are crucial for establishing the relationship of the invariants with the Casson invariant (Mu1, Mu2) and with the perturbative invariants or Ohtsuki series (Oh1, Oh2, Le). Quantum invariants fit into Topological Quantum Field Theories (TQFT). This means in particular that there are representations of mapping class groups associ- ated with them. (Actually the representations are usually only projective-linear; equivalently, one has to consider certain central extensions of mapping class groups here.) If a 3-manifold M is presented as a Heegaard splitting where two handlebod- ies are glued together by a diffeomorphism ϕ along their boundary, the quantum invariant of M can be recovered from the representation of ϕ on the TQFT-vector space V (Σ) associated to the boundary surface Σ. The TQFT-representations are finite-dimensional and can be defined over a finite extension of the cyclotomic number field Q(q), where the quantum parameter

Abstract: Let be the complex Banach algebra of all bounded linear operators on a complex Banach space E For n-tuples and of operators on E, let R A,B denote the operator on defined by For we put In this note, we prove that where V(·) is the joint spatial numerical range, W 0(·) is the algebraic numerical range and J is a norm ideal of We shall show that this inclusion becomes an equality when R A,B is taken to be a derivation. Also, we deduce that for and J is a norm ideal of , where w(·) is the numerical radius. On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound holds, if one of the following two conditions is satisfied: J is a standard operator algebra of and A,B∈ J. J is a norm ideal of and

Abstract: A proof is given to show that the operator Hilbert space OH does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge’s recent result, which admits such an embedding in the non-semi-finite case. In remarkable recent work [5], Marius Junge proves that the operator Hilbert space OH (from [8]; see also [10]) embeds completely isomorphically into the predual M∗ of a von Neumann algebra M which is of type III; thus this algebra M is not semi-finite. In this paper, we show that no such embedding can exist when M is semi-finite. The results that we have just stated all belong to the currently very active field of ‘operator spaces’, for which we refer the reader to the monographs [2, 11]. We merely recall a few basic facts, relevant to the present paper. An operator space is a Banach space, given together with an isometric embedding E ⊂ B(H )i nto the algebra B(H) of all bounded operators on a Hilbert space H. Using this embedding, we equip the space Mn(E) (consisting of the n × n matrices with entries in E) with the norm induced by the space Mn(B(H)), naturally identified isometrically with B(H ⊕···⊕ H).