On Lie derivations of Lie ideals of prime algebras

LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA _c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt ₁₄, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A _c such that for allx∈R. Our result solves an old problem of Herstein.     

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If and are saturated formations then is defined to be the class of all soluble groups whose belong to . In general is a formation, but need not be a saturated formation. Here the smallest saturated formation containing is studied.     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.     

On the maximal dimension of a completely entangled subspace for finite level quantum systems

LetH _ibe a finite dimensional complex Hilbert space of dimensiond _i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂ is said to becompletely entangled if it has no non-zero product vector of the formu ₁⊗u ₂ ⊗ ... ⊗u _k with u_i inH _i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Pseudo algebraically closed fields over rings

We prove that for almost allσ ∈G ℚ the field has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→ there exists a point a ∈ such thatϕ(a) ∈ ℤ^r. We then say that is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

Reality properties of conjugacy classes in algebraic groups

Let be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra so that for every 1 < p < 2, the Haagerup L ^p -space associated with embeds isomorphically into . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if is a semi-finite von Neumann algebra then every reflexive subspace of embeds isomorphically into L ^r ( ) for some r > 1. Dedicated to Professor H. P. Rosenthal on the occasion of his sixty-fifth birthday Research partially supported by NSF grant DMS-0456781.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

An example of pointwise non-convergence of iterated conditional expectation operators

Given ∈, we construct a sequence , … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on , then , then , Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈. The sequence of functions also will fail to converge in the . This contrasts with the long-open conjecture that if all the come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in . J. L. King was partially supported by NSF grant DMS-9112595.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

Completion of restricted Lie algebras

We study pro-‘finite dimensional finite exponent’ completions of restricted Lie algebras over finite fields of characteristicp. These compact Hausdorff topological restricted Lie algebras, called pro- restricted Lie algebras, are the restricted Lie-theoretic analogues of pro-p groups. A structure theory for pro- restricted Lie algebras with finite rank is developed. In particular, the centre of such a Lie algebra is shown to be open. As an application we examinep-adic analytic pro-p groups in terms of their associated pro- restricted Lie algebras. Supported by NSERC of Canada.     

On the singularities of nilpotent orbits

Let be a nilpotent orbit of the adjoint action of a complex connected semi-simple Lie group on its Lie algebra. We prove that the normalization of the closure of is Gorenstein and has rational singularities.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.