Effect Algebras with the Riesz Decomposition Property and AF C*-Algebras

Relations between effect algebras with Riesz decomposition properties and AF C^*-algebras are studied. The well-known one-one correspondence between countable MV-algebras and unital AF C^*-algebras whose Murray-von Neumann order is a lattice is extended to any unital AF C^* algebras and some more general effect algebras having the Riesz decomposition property. One-one correspondence between tracial states on AF C^*-algebras and states on the corresponding effect algebras is proved. In particular, pure (faithful) tracial states correspond to extremal (faithful) states on corresponding effect algebras.     

Coherent States with Complex Functions

The canonical coherent states are infinite series in powers of a complex number z. We present classes of coherent states by replacing this complex number z by other choices, namely, iterates of a complex function, higher functions, and elementary functions. Further, we show that some of these classes do not furnish generalized oscillator algebras in the natural way. A reproducing kernel Hilbert space is discussed to each class of coherent states.     

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.     

Weighted Circle Actions on the Heegaard Quantum Sphere

Weighted circle actions on the quantum Heeqaard 3-sphere are considered. The fixed point algebras, termed quantum weighted Heegaard spheres, and their representations are classified and described on algebraic and topological levels. On the algebraic side, coordinate algebras of quantum weighted Heegaard spheres are interpreted as generalised Weyl algebras, quantum principal circle bundles and Fredholm modules over them are constructed, and the associated line bundles are shown to be non-trivial by an explicit calculation of their Chern numbers. On the topological side, the C*-algebras of continuous functions on quantum weighted Heegaard spheres are described and their K-groups are calculated.     

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.     

Mean entropy of states in classical statistical mechanics

The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with theKolmogorov-Sinai invariant of ergodic theory.     

Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita’s theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan ?-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.     

Distances in finite spaces from noncommutative geometry

Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in finite commutative cases corresponding to a metric on a finite set, and also compute explicitly some distances associated to commutative or noncommutative algebras.     

{Quantum Algebras and q-Special Functions Related to Coherent States Maps of the Disc

The quantum algebras generated by the coherent states maps of the disc are investigated. It is shown that the analytic realization of these algebras leads to a generalized analysis which includes standard analysis as well as q-analysis. The applications of the analysis to star-product quantizations and q-special functions theory are given. Among others the meromorphic continuation of the generalized basic hypergeometric series is found and a reproducing measure is constructed, when the series is treated as a reproducing kernel. Received: 4 April 1996 / Accepted: 29 June 1997     

Boundary Hamiltonian Theory for Gapped Topological Orders

We report our systematic construction of the lattice Hamiltonian model of topological orders on open surfaces,with explicit boundary terms. We do this mainly for the Levin-Wen string-net model. The full Hamiltonian in our approach yields a topologically protected, gapped energy spectrum, with the corresponding wave functions robust under topology-preserving transformations of the lattice of the system. We explicitly present the wavefunctions of the ground states and boundary elementary excitations. The creation and hopping operators of boundary quasi-particles are constructed. It is found that given a bulk topological order, the gapped boundary conditions are classified by Frobenius algebras in its input data. Emergent topological properties of the ground states and boundary excitations are characterized by(bi-) modules over Frobenius algebras.     

Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.     

On States and State Operators on Certain Basic Algebras

The paper is based on the authors’ talks given at the 11th IQSA Meeting in Cagliari; it deals with states and state operators (internal states) on basic algebras which are a generalization of MV-algebras and orthomodular lattices, also including lattice effect algebras. On the one hand, the paper is a survey of some previous results on states and state operators on commutative basic algebras, and on the other one, an extension of these results to the much larger class of basic algebras where the addition ⊕ distributes over the lattice meet ∧.     

Affine Maczyński logics on compact convex sets of states

Sets of affine functions satisfying Maczyński orthogonality postulate and defined on compact convex sets of states are examined. Relations between affine Maski logics and Boolean algebras when the set of states is a Bauer simplex (classical mechanics, some models of nonlinear quantum mechanics) are studied. It is shown that an affine Maczyński logic defined on a Bauer simplex is a Boolean algebra if it is a sublattice of a lattice consisting of all bounded affine functions defined on the simplex.     

States on Clifford algebras

We study states on Clifford algebras from the point of view of C*-algebras. A criterium is given under which the odd-point functions vanish. A particular set of states, called quasi-free states is extensively studied and explicit representations are given; as an application we give an approximate calculation of the ground state of a Fermion system.On leave from Matematisk Institut, University of Aarhus, Denmark.Aangesteld navorser van het Belgisch N. F. W. O. On leave from University of Louvain, Belgium.     

Meromorphic continuation approach to noncommutative geometry

Following an idea of Nigel Higson, we develop a method for proving the existence of a meromorphic continuation for some spectral zeta functions. The method is based on algebras of generalized differential operators. The main theorem states, under some conditions, the existence of a meromorphic continuation, a localization of the poles in supports of arithmetic sequences and an upper bound of their order. We give an application in relation to a class of nilpotent Lie algebras.     

Generalized Barut-Girardello Coherent States for Mixed States with Arbitrary Distribution

In the paper we examine some properties of the generalized coherent states of the Barut-Girardello kind. These states are defined as eigenstates of a generalized lowering operator and they are strongly dependent on the structure constants. Besides the pure coherent states we focused our attention on the mixed states one, which are characterized by different probability distributions. As some examples we consider the thermal canonical distribution and the Poisson distribution functions. We calculate for these cases the Husimi’s Q and quasi-probability P-distribution functions.     

An Algebraic Characterization of Vacuum States in Minkowski Space. II. Continuity Aspects

An algebraic characterization of vacuum states in Minkowski space is given which relies on recently proposed conditions of geometric modular action and modular stability for algebras of observables associated with wedge-shaped regions. In contrast to previous work, continuity properties of these algebras are not assumed but derived from their inclusion structure. Moreover, a unique continuous unitary representation of spacetime translations is constructed from these data. Thus, the dynamics of relativistic quantum systems in Minkowski space is encoded in the observables and state and requires no prior assumption about any action of the spacetime symmetry group upon these quantities.     

Congruences and States on Pseudoeffect Algebras

We study congruences on pseudoeffect algebras, which were recently introduced as a non-commutative generalization of effect algebras. We introduce ideals for these algebras and give a sufficient condition for an ideal to determine a congruence. Furthermore, states on pseudoeffect algebras are considered. It is shown that any interval pseudoeffect algebra maps homomorphically into an effect algebra whose states are in a one-to-one correspondence to the states of the original algebra.     

The Lattice and Simplex Structure of States on Pseudo Effect Algebras

We study states, measures, and signed measures on pseudo effect algebras with some version of the Riesz Decomposition Property (RDP). We show that the set of all Jordan signed measures is always an Abelian Dedekind complete ℓ-group. Therefore, the state space of a pseudo effect algebra with RDP is either empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow to represent states on pseudo effect algebras by standard integrals.     

Joint Extension of States of Subsystems for a CAR System

 The problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a Fermion system, where a novel feature is non-commutativity between algebras of subsystems. For an arbitrary (finite or infinite) number of given subsystems, a product state extension is shown to exist if and only if all states of subsystems except at most one are even (with respect to the Fermion number). If the states of all subsystems are pure, then the same condition is shown to be necessary and sufficient for the existence of any joint extension. If the condition holds, the unique product state extension is the only joint extension. For a pair of subsystems, with one of the given subsystem states pure, a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions (unique for almost all cases) are given. For a pair of subsystems with non-pure subsystem states, some classes of examples of joint extensions are given where non-uniqueness of joint extensions prevails. Received: 17 May 2002 / Accepted: 16 January 2003 Published online: 17 April 2003 Communicated by D. Buchholz and K.Fredenhagen