Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

The Ideal Structure of Toeplitz Algebras

We investigate the ideal structure of the Toeplitz algebra of a totally ordered abelian group . We show that the primitive ideals of are parametrised by the disjoint union of the duals of the order ideals of , and identify the hull-kernel topology on when the chain of orderideals in is isomorphic to a subset of      

C*-Modular Vector States

Let be a C*-algebra and X a Hilbert C* -module. If is a projection, let be the p-sphere of X. For φ a state of with support p in and consider the modular vector state φ_x of given by The spheres provide fibrations

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Let E be a non empty set, let P : = E × E, := {x × E|x ∈ E}, := {E × x|x ∈ E}, and := {C ∈ 2 ^P |∀X ∈ : |C ∩ X| = 1} and let . Then the quadruple resp. is called chain structure resp. maximal chain structure. We consider the maximal chain structure as an envelope of the chain structure . Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms , , , related to a maximal chain structure . The set of all chains can be turned in a group such that the subgroup of generated by the left-, by the right-translations and by ι the inverse map of is isomorphic to (cf. (2.14)).     

Complete Positivity of Elementary Operators on C*-Algebras

Let be a C*-algebra. We obtain some conditions that are equivalent to the statement that every n-positive elementary operator on is completely positive.     

On an Algebra of Toeplitz Operators With Piecewise Continuous Symbols

Let and be a finite collection of smooth curves in D. Given k points consider the family of all bounded and continuous functions on with finite limits at and radial limits at z_k. We study the Toeplitz operator algebra corresponding to M_r and we prove that its Calkin algebra is isomorphic to the algebra of all continuous functions on some compact set. This fact implies that the commutator of two Toeplitz operators with this kind of symbols is compact. We also prove that the semi-commutator of such Toeplitz operators is not compact, in general.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

Let be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x ₁,x ₂),x ₃), ((x ₄,x ₅),x ₆)), (x ₇,x ₈)) = 0 (((x ₁ x ₂ · x ₃)(x ₄ x ₅ · x ₆))(x ₇ x ₈) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety ( , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety has the cardinality of the continuum and that there are algebras in with undecidable word problem.     

Homological Dimensions of the Algebra Formed by Entire Functions of Elements of a Nilpotent Lie Algebra

This note deals with homological characteristics of algebras of holomorphic functions of noncommuting variables generated by a finite-dimensional nilpotent Lie algebra . It is proved that the embedding of the universal enveloping algebra of into its Arens–Michael hull is an absolute localization in the sense of Taylor provided that      

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

Varieties Defined by Permutations

We continue to study interrelations between permutative varieties and the cyclic varieties defined by cycles of the form . A criterion is given determining whether a cyclic variety is interpretable in . For a permutation without fixed elements, it is stated that a set of primes for which is interpretable in in the lattice is finite. It is also proved that for distinct primes , the Helly number of a type in coincides with dimension of the dual type and equals .     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

A family of transitive modular Lie superalgebras with depth one

The embedding theorem is established for Z-graded transitive modular Lie superalgebras g=(?)satisfying the conditions: (i)g0(?)(g-1)and g0-module g-1 is isomorphic to the natural(?)(g-1)-module; (ii)dim g1=2/3n(2n~2 1),where n=1/2dim g-1. In particular,it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras.The restricted Lie superalgebras are also considered.     

Factorizations of One-Generated Composition Formations

A non-empty formation of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group G having a normal subgroup N such that belongs to . An intersection of all composition formations containing a given group G is denoted cformG. Conditions are described under which has the form , where .     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

Symmetries of Systems of the Hyperbolic Riccati Type

Let be a Kac–Moody algebra, U(x,y) be a function defined in , and a be a constant element of . We prove that the equation U _xy = [[U,a],U _x] has two symmetry hierarchies connected by a gauge transformation. In particular, the well-known Konno equation appears in the case of the algebra . The corresponding symmetry hierarchies contain the nonlinear Schrödinger and the Heisenberg magnet equations.