Spectral Truncations in Noncommutative Geometry and Operator Systems

Some recent research on the tensor products of operator systems and ensuing nuclearity properties in this setting raised many stability problems. In this paper we examine the preservation of these nuclearity properties including exactness, local liftability and the double commutant expectation property under basic algebraic operations such as quotient, duality, coproducts and tensor products. We show that, in the finite dimensional case, exactness and lifting property are dual pairs, that is, an operator system $S$ is exact if and only if the dual operator system $S^d$ has the lifting property. Moreover, the lifting property is preserved under quotients by null subspaces. Again in the finite dimensional case we prove that every operator system has the k-lifting property in the sense that whenever $f:S -> A/I$ is a unital and completely positive map, where A is a C*-algebra and I is an ideal, then $f$ possess a unital k-positive lift on A, for every k. This property provides a novel proof of a classical result of Smith and Ward on the preservation of matricial numerical ranges of an operator. The Kirchberg conjecture naturally falls into this context. We show that the Kirchberg conjecture is equivalent to the statement that the five dimensional universal operator system generated by two contraction ($S_2$) has the double commutant expectation property. In addition to this we give several equivalent statements to this conjecture regarding the preservation of various nuclearity properties under basic algebraic operations. We show that the Smith Ward problem is equivalent to the statement that every three dimensional operator system has the lifting property (or exactness). If we suppose that both the Kirchberg conjecture and the Smith Ward problem have an affirmative answer then this implies that every three dimensional operator system is C*-nuclear.

We study the duals of a certain class of finite-dimensional operator systems, namely the class of operator systems associated to tolerance relations on finite sets or equivalently the class of operator systems that are associated with graphs. In the case where the graphs associated with these operator systems are chordal we are able to find concrete realizations of their duals as sitting inside of finite-dimensional C∗-algebras. We then use these concrete realizations to compute the C∗-envelopes, propagation numbers and extremal rays of these duals in the chordal case. Finally, we exemplify our results by applying them to operator systems of band matrices.

Role of optimal sized dual energy storage based microgrid for ancillary service support to utility grid

Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: relativistic space-time symmetries Poincare group as a classical deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

We give an introduction to the basic notions of noncommutative geometry including the calculus of infinitesimals with operators, cyclic cohomology and the local index formula. We also explain in details how the infinitesimal calculus based on operators gives a natural home for the infinitesimal line element ds of geometry and leads one to the basic notion of spectral triple, which is the basic paradigm of noncommutative geometry. In order to illustrate these general concepts we then analyse the noncommutative space underlying the quantum group SU q (2) from this spectral point of view, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic cocycle giving the index formula. This specific example allows to illustrate the general notion of locality in noncommutative geometry. The formulas computing the residue are ”local”. Locality by stripping all the expressions from irrelevant details makes them easily computable. The original Chern character is non-local and the cochain whose coboundary is the difference between the original Chern character and the local one is much harder to compute than the local cochains. It is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. The key feature of this spectral triple is its equivariance, i.e. the SU q (2)-symmetry. We explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries and relate this notion to previous work.KeywordsDirac OperatorQuantum GroupPseudodifferential OperatorNoncommutative GeometryChern CharacterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we define a class of stable equivalences, namely, the stable equivalences of adjoint type, and study the Hochschild cohomology groups of algebras that are linked by a stable equivalence of adjoint type. This notion of adjoint type is a special case of Morita type, covers the stable equivalence of Morita type for self-injective algebras, and thus includes the case where Broué's conjecture was made (see for instance [Broué M.: Equivalences of blocks of group algberas. In: Finite dimensional algebras and related topics (V. Dlab and L. L. Scott eds.). Kluwer, 1994, 1–26]). The main results in this paper are: Let A and B be two artin k-algebras such that A and B are projective over k, and let Hn(A) and Hn(B) be the n-th Hochschild cohomology groups of A and B, respectively. (1) If A and B are stably equivalent of adjoint type, then Hn(A) ≃ Hn(B) for all n ≥ 1. (2) If A and B are stably equivalent of Morita type, then the absolute values of Cartan determinants of A and B are equal. In particular, two cellular algebras over a field have the same Cartan determinant if they are stably equivalent of Morita type.

The present thesis focuses on the study of the mathematical structures underlying input/output logic. Building on insights from Abstract Algebraic Logic and on the recent introduction of subordination algebras and related structures as a semantic environment for input/output logic, the present thesis establishes a strong and systematic bridge between areas of logic which have been investigated for decades independently of each other, and by doing so, it opens new possibilities of interpretations and areas of applications for these formal frameworks. Motivated by a research program investigating ‘logic at work’ rather than ‘logic in isolation’, input/output logic has been introduced for modelling the interaction between logical inferences and other agency-related notions such as conditional obligations, goals, ideals, preferences, actions, and beliefs, in the context of the formalization of normative systems in philosophical logic and AI, but also in connection and combination with a very diverse range of issues, also relevant to present-day AI, spanning from formal argumentation, to causal reasoning and non-monotonic reasoning. The first contribution of this thesis is a generalization of the framework of input/output logic from classical propositional logic to selfextensional logics, a large class of logics which prominently includes those logical systems the consequence relation of which corresponds to a partial order on the algebras canonically associated with it. The various notions of permission systems, namely, negative permission, positive static permission and dynamic permission, are generalized and studied uniformly in the context of selfextensional logics, both in themselves, and in connection with normative systems. In the same context, the notion of dual negative permission system is introduced and studied. The second contribution of this thesis is a systematic study of the algebraic counterparts of normative, permission, and dual permission systems based on the algebras of the classes canonically associated with (fully) selfextensional logics. In particular, the proof of the main characterization result of the previous chapter is refined and streamlined, and further extended to characterize properties of various notions of permission systems also in relation to normative systems, by means of modal axioms in an expanded slanted signature which includes negative modal operators (intuitively representing prohibitions). Moreover, these characterization results are then applied to resolve different issues: from the logical characterizations of output operators for both normative and permission systems, to the dual characterizations of conditions on subordination algebra (resp. precontact algebras) and on their associated spaces; the algebraization of positive static permissions; the modal characterization of the notion of cross-coherence; the clarification of the relation between certain conditions on positive bi-subordination lattices and Dunn’s axioms for positive modal logic. The third contribution of this thesis generalizes the modal characterizations, achieved in the previous chapter, of finitely many conditions on normative and permission systems, as well as on subordination algebras and related structures, to those conditions corresponding to the infinite class of clopen-analytic inequalities. Conversely, Kracht (first-order) formulas are introduced as the class of those first order conditions on algebras with subordination, precontact, and dual precontact relations which correspond to clopen-analytic axioms; finally, as an application of these results, Celani’s dual characterization results between subordination lattices and subordination spaces is generalized to the environment of algebras with relations based on distributive lattices and the infinite class of inductive Kracht formulas.

In light microscopy it is natural for everybody to use manual or automatic toolsets like tweezers, knives, hooks, probes and several different measurement tools. Without such handling, manipulation and manufacturing tools many present-day products and methods would not exist. No wristwatch, no in vitro fertilization, no mini-gearbox, just to mention a few. The operators of SEM, FIB or Dual Beam systems generally work without toolsets and call it natural, although the wavelength limit of light is no physical boundary for using such tools. It can be imagined how technology would be pushed when a SEM/FIB Workbench reaches the same degree of practicability and utilization as toolsets for light microscopes.

The (2[Formula: see text]+[Formula: see text]1)-dimensional Klein–Gordon oscillator under a magnetic field in the presence of a minimal length in the noncommutative (NC) space is analyzed via the momentum space representation. Energy eigenvalue of the system is obtained by employing the Jacobi polynomials. In further steps, the special cases are discussed and the corresponding numerical results are depicted, respectively.

We give a review of our joint work on strict deformation of BHTZ 2+1 black holes \cite{BRS02,BDHRS03}. However some results presented here are not published elsewhere, and an effort is made for enlightening the instrinsical aspect of the constructions. This shows in particular that the three dimensional case treated here could be generalized to an anti-de Sitter space of arbitrary dimension provided one disposes of a universal deformation formula for the actions of a parabolic subgroup of its isometry group. Comment: 10 pages, based on a talk given by P.B., to appear in the proceedings of the workshop `Noncommutative Geometry and Physics 2004' (Feb. 2004, Keio University, Japan) (World Scientific)

This collection of invited mathematical papers by a list of distinguished mathematicians is an outgrowth of the scientific activities at the Center for Geometry and Mathematical Physics of Penn State University. The articles in this text present results or discuss perspectives on work that should be of interest to researchers and graduate students working in symplectic geometry and geometric quantization, deformation quantization, non-commutative geometry and index theory, quantum groups, holomorphic algebraic geometry and moduli spaces, quantum cohomology, algebraic groups and invariant theory, and characteristic classes.

In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the M theory on noncommutative tori. This turns out to be an interesting field of applications, since the space [Formula: see text] of the equivalence classes of the vector unitary irreducible representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the algebraic integration is equivalent to integrate over [Formula: see text]. However, its very definition is related only at the structural properties of the group algebra, therefore it is well defined also in the projective case, where the space [Formula: see text] has no classical meaning. This allows a generalization of the usual group harmonic analysis. Particular attention is given to Abelian groups, which are the relevant ones in the compactification problem, since it is possible, from the previous results, to establish a simple generalization of the ordinary calculus to the associated noncommutative spaces.

Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics. In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations. We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the h- and q-derivatives, respectively. In these various examples, some points turn out to be at infinite distance. We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance. On the circle, this provides an explicit example of computation of the Wasserstein distance.

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

Model-independent low momentum nucleon interaction from phase shift equivalence