S-Dominating Effect Algebras

A special type of effect algebra called anS-dominating effect algebra is introduced. It is shownthat an S-dominating effect algebra P has a naturallydefined Brouwer-complementation that gives P thestructure of a Brouwer–Zadeh poset. This enables usto prove that the sharp elements of P form anorthomodular lattice. We then show that a standardHilbert space effect algebra is S-dominating. Weconclude that S-dominating effect algebras may be usefulabstract models for sets of quantum effects in physicalsystems.     

Structure of the Time Projection for Stopping Times in von Neumann Algebras

We give an explicit formula for the time projection in an arbitrary von Neumann algebra from which all its basic properties can be easily derived. The analysis of the situation when this time projection is a conditional expectation is also performed.     

States and Structure of von Neumann Algebras

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory.     

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lüders—von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lüders—von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.     

Pre-Poisson Algebras

A definition of pre-Poisson algebras is proposed, combining structures of pre-Lie and zinbiel algebra on the same vector space. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra by passing to the corresponding Lie and commutative products. Analogs of basic constructions of Poisson algebras (through deformations of commutative algebras, or from filtered algebras whose associated graded algebra is commutative) are shown to hold for pre-Poisson algebras. The Koszul dual of pre-Poisson algebras is described. It is explained how one may associate a pre-Poisson algebra to any Poison algebra equipped with a Baxter operator, and a dual pre-Poisson algebra to any Poisson algebra equipped with an averaging operator. Examples of this construction are given. It is shown that the free zinbiel algebra (the shuffle algebra) on a pre-Lie algebra is a pre-Poisson algebra. A connection between the graded version of this result and the classical Yang–Baxter equation is discussed.     

Algebras of Random Operators Associated to Delone Dynamical Systems

We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C ^*-subalgebra to discuss a Shubin trace formula.     

Groupoids,von Neumann Algebras and the Integrated Density of States

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrödinger operators on manifolds, quantum percolation and quasi–crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.     

Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition

Quantum bialgebras derivable from U _q (sl ₂) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition. Dedicated to the memory of our colleague Leonid L. Vaksman (1951–2007) On leave of absence from: TheoryGroup, Nuclear Physics Laboratory,V.N.Karazin Kharkov National University, Svoboda Sq. 4, Kharkov 61077, Ukraine. E-mail: sduplij@gmail.com;     

Quantum Kac– Algebras and Vertex Representations

We introduce an affinization of the quantum Kac–Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac–Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial indentity about Hall–Littlewood polynomials.     

Inhomogeneous Yang-Mills Algebras

We determine all inhomogeneous Yang–Mills algebras and super Yang–Mills algebras which are Koszul. Following a recent proposal, a non-homogeneous algebra is said to be Koszul if the homogeneous part is Koszul and if the PBW property holds. In this letter, the homogeneous parts are the Yang–Mills algebra and the super Yang–Mills algebra.     

On the FRTS Approach to Quantized Current Algebras

We study the possibility to establish L-operator's formalism by Faddeev–Reshetikhin–Takhtajan–Semenov-Tian-Shansky (FRST) for quantized current algebras, that is, for quantum affine algebras in the new realization by V. Drinfeld with the corresponding Hopf algebra structure and for their Yangian counterpart. We establish this formalism using the twisting procedure by Tolstoy and the second author and explain the problems which on FRST approach encounters for quantized current algebras. We also show that, for the case of U_q(l_n), entries of the L-operators of the FRTS type give the Drinfeld current operators for the nonsimple roots, which we discovered recently. As an application, we deduce the commutation relations between these current operators for U_q(l₃).     

Galilean-Covariant Clifford Algebras in the Phase-Space Representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.     

Spectral Lattices

Spectral orthomorphisms between the spectral lattices of JBW algebras which preserve the scales extend to Jordan homomorphisms for a large class of algebras. Spectral lattice homomorphism is automatically a σ-lattice homomorphism. The range projection map is, up to a Jordan homomorphism, the only natural map from the spectral lattice onto the projection lattice. Continuity of the range projection determines finiteness of the algebra in Murray–von Neumann comparison theory.     

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.     

Some Results on Novikov–Poisson Algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. A Novikov–Poisson algebra is a Novikov algebra with a compatible commutative associative algebraic structure, which was introduced to construct the tensor product of two Novikov algebras. In this paper, we commence a study of finite-dimensional Novikov–Poisson algebras. We show the commutative associative operation in a Novikov–Poisson algebra is a compatible global deformation of the associated Novikov algebra. We also discuss how to classify Novikov–Poisson algebras. And as an example, we give the classification of 2-dimensional Novikov–Poisson algebras.     

Remarks on Effect Algebras

Erik M. Alfsen and Frederic W. Shultz had recently developed the characterisation of state spaces of operator algebras. It established full equivalence (in the mathematical sense) between the Heisenberg and the Schr?dinger picture, i.e. given a physical system we are able to construct its state space out of its observables as well as to construct algebra of observables from its state space. As an underlying mathematical structure they used the theory of duality of ordered linear spaces and obtained results are valid for various types of operator algebras (namely C ^*, von Neumann, JB and JBW algebras). Here, we show that the language they developed also admits a representation of an effect algebra.     

States on projection logics of von Neumann algebras

We summarize recent results concerning states on projection lattices of von Neumann algebras. In particular, we present an analysis of the Jauch-Piron property in the von Neumann algebra setting.     

Tensor Product of Distributive Sequential Effect Algebras and Product Effect Algebras

A distributive sequential effect algebra (DSEA) is an effect algebra on which a distributive sequential product with natural properties is defined. We define the tensor product of two arbitrary DSEA’s and we give a necessary and sufficient condition for it to exist. As a corollary we obtain the result (see Gudder, S. in Math. Slovaca 54:1–11, 2004, to appear) that the tensor product of a pair of commutative sequential effect algebras exists if and only if they admit a bimorphism. We further obtain a similar result for the tensor product of a pair of product effect algebras.     

Some Infinite-Dimensional Algebras Arising in Spin Systems and in Particle Physics and their Grand Algebra

We indicate similarities in the structure of two types of infinite-dimensional algebras, one introduced 28 years ago in connection with the mass problem of elementary particles and the other seven years ago in connection with spin systems (XY models). We show that these algebras can be considered as representations of a single Grand Algebra, the enveloping algebra of an affine Kac–Moody algebra built on the Poincaré Lie algebra. As an associative and coassociative bialgebra of operators, the latter representation of the grand algebra is a preferred nontrivial deformation of the Ising case bialgebra.