Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Automatic structures,rational growth,and geometrically finite hyperbolic groups

Summary We show that the set of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets over all maximal parabolic subgroupsP. The set of equivalence classes of biautomatic structures onG is isomorphic to the product of the sets over the cusps (conjugacy classes of maximal parabolic subgroups) ofG. Each maximal parabolicP is a virtually abelian group, so and were computed in [NS1].We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics forG is regular. Moreover, the growth function ofG with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Oblatum 14-VI-1993 & 4-I-1994Both authors acknowledge support from the NSF for this research.     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

A new product of algebras and a type reduction theorem

A construction is defined which associates, to every algebra of a fixed but arbitrary finite similarity type, a groupoidF . The identities ofF are finitely based if and only if those of are, andF is finite if and only if is finite. Up to isomorphism,F has the same endomorphism monoid and subalgebra lattice as , but the congruence lattice ofF is the result of adjoining a new 1 to the congruence lattice of .F is functorial, preserves the satisfaction (and the non-satisfaction) of most Mal'cev conditions, and produces, by composition with the operation of forming the generated variety, an isomorphism of the lattice of varieties of fixed type to an interval in the lattice of varieties of groupoids.The construction makes use of a new product operation, applicable to two algebras of differing similarity types, which is introduced and studied in this paper.Research supported by National Science Foundation grant MCS-8103455.Presented by K. A. Baker.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Commutator-finite orthomodular lattices

The class of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: contains the block-finite orthomodular lattices. Every irreducible element of is simple. Every element of is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of which are modular, or are M-symmetric with at least one atom, have height two or less.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

The closure of unitary orbit for strongly irreducible operators in a class of nest algebras

A linear operatorT L(H) is called a strongly irreducible, if there is no non-trivial idempotent linear operator commuting withT. In this paper, denote the set of all strongly irreducible operators by (SI). Let be a nest with infinite dimensional atoms, be the nest algebra associated with and be the closure of , then the following result is proved

Nilpotent shifts on manifolds

On the lattice of manifolds of all algebras L we study the operator of nilpotent closure , where is a nilpotent manifold of -algebras. With a given system of identities defining, we construct a system ^*, giving the manifold It is proved that if does not contain , then the lattice of submanifolds of is the double of the lattice of submanifolds of. We describe the free and subdirect indecomposable manifolds of algebras . Let and A be adense retract of B. We denote by (B) the lattice of congruences on B. The theorem is proved: (B) is a complemented lattice if and only if (A) is a complemented lattice.Translated from Matematicheskie Zametki, Vol. 14, No. 5, pp. 703–712, November, 1973.     

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).     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

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.     

Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

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.