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 .     

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     

On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.     

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.     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

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 .     

Repleteness and regular lattice measures

Let X be an abstract set and L a lattice of subsets of X. To each lattice regular measure µ, we associate two induced measures and on suitable lattices of the Wallman space IR(L) and another measure µ on the spaceIR(L). We will investigate the reflection of smoothness properties of p onto , and µ; and try to set some new criterion for repleteness and measure repleteness.     

The space of minimal prime ideals in a 0-distributive semilattice

LetS be a 0-distributive semilattice and be its minimal spectrum. It is shown that is Hausdorff. The compactness of has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.     

The Martin boundary and completion of Markov chains

Summary In this paper we treat a time-symmetrical Martin boundary theory for continuous parameter Markov chains. This is done by reversing the time sense of a Markov chainX _t in such a way as to obtain a dual Markov chain , and considering the two chains together. Various relations between the Martin exit boundaries and of these processes are studied. The exit boundary of , is in a sense an entrance boundary forX _t and vice versa. After a natural identification of certain points in and one can topologizeI in such a way thatboth X _t and have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.Research supported in part at Stanford University, Stanford, California under AFOSR 0049.     

Spectral pictures of Aluthge transforms of operators

In this paper we continue our study, begun in [12], of the relationships between an arbitrary operatorT on Hilbert space and its Aluthge transform . In particular, we show that in most cases the spectral picture ofT coincides with that of , and we obtain some interesting connections betweenT and as a consequence.     

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.     

Computing on-line the lattice of maximal antichains of posets

We consider the on-line computation of the lattice of maximal antichains of a finite poset . This on-line computation satisfies what we call the linear extension hypothesis: the new incoming vertex is always maximal in the current subposet of . In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of , where |P| is the number of elements of the poset, , |MA(P)| is the number of maximal antichains of and (P) is the width of . This is more efficient than the best off-line algorithms known so far.     

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

Invariant differental forms on compact nilmanifolds

Let N=G/ be a compact nilmanifold, G a connected, simply connected, nilpotent Lie group with its discrete subgroup and Lie algebra . Let I* ( ) denote the invariant differential forms on .If I* ( ) H* ( ) is an injective map, then G is abelian and N is a torus. Furthermore, N has a formal minimal model. If N is an even-dimensional compact nilmanifold, it has a Kähler structure and invariant symplectic structure if and only if I* ( ) H* ( ) is injective.     

Complétude des métriques lorentziennes de\mathbb{T}^2 et difféomorphismes du cercle

Let and the foliations by the null geodesics of some lorentzian metricg on the torus . We analyse how geodesic completeness properties ofg are related to the dynamics of and .     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

The modulus of Rudin-Carleson extensions

We prove the followingTheorem. LetF be a closed subset of the unit circleT which has Lebesgue measure zero. Suppose thatp is a smooth positive function onT. GivenfC(F) which satisfies|f(s)|p(s) (sF) and a neighbourhoodU ofF there is an extension off in the disc algebra such that and .     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Bemerkungen zur Deformationstheorie nichtrationaler Singularitäten

Let be a 1-convex holomorphic mapping between complex spaces resp.S, and let be the blowingdown factorization of over S. We prove in part 1 of the present note: The fiber ^–1(s₀) over a point s₀S is the Remmert quotient of if and only if every holomorphic function on (defined in a neighborhood of the exceptional subvariety of that fiber) can be extended holomorphically to . This is true, for instance, in the case: flat, S reduced at s₀ and dim , =const for all sS. In part 2, we use this result to obtain the following: For any Riemann surface R with genus g2 there exists a 2-dimensional normal complex analytic singularity X such that the minimal resolution of X contains R as exceptional subvariety, and has a deformation over the unit disc S={|s|<1} which can not be blown down to a deformation of X.