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.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

On equivariant Kasparov theory and Spanier-Whitehead duality

Suppose thatG is a second countable compact Lie group and thatA andB are commutativeG-C*-algebras. Then the Kasparov groupKK _* ^G (A, B) is a bifunctor onG-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of -G-C*-algebras. As a consequence, we show that if (X, x ₀) is a based separable compact metricG-ENR (such as a smooth compactG-manifold) and (Y, y ₀) is a based countableG-CW-complex then there is a natural isomorphism

Characterizations of logA≥logB and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB ^r X^1–r A ^r XB ^r holds for r [0, 1]. In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logAlogB) by a norm inequality.Secondly, we shall study the condition under which , where is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=T) via Aluthge transformation.     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

Duality for contravariant functors on Banach spaces

The following notion of duality is studied: IfG is a contravariant functor on Ban, then . We derive the following results: The^x-reflexive functors are exactly the maximal subfunctors ofH(.,A) with reflexiveA, G ^x is^x-reflexive if and only ifG(I) is a reflexive Banach space,G ^x=(G_e)^x, andG ^x(X)=G_e(X) ifX has the metric approximation property. the last result has the consequence that for the tensor product of functors holds, ifX has the m. A. P. Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday     

Sur le problème des marges

Summary LetX be a set,A an algebra of subsets ofX, m andM two mappings fromA to . Then there exists a finitely additive measure onA such thatmM if and only if for all the sequences (A ₁, ...,A _p ) and (B ₁, ...,B _q ) inA such that the inequality is satisfied. This simple condition permits us to deduce and generalize many previous results relating to the marginal problem.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Bezout Rings,Polynomials, and Distributivity

Let A be a ring, be an injective endomorphism of A, and let be the right skew polynomial ring. If all right annihilator ideals of A are ideals, then R is a right Bezout ring is a right Rickartian right Bezout ring, (e)=e for every central idempotent eA, and the element (a) is invertible in A for every regular aA. If A is strongly regular and n 2, then R/x ⁿ R is a right Bezout ring R/x ⁿ R is a right distributive ring R/x ⁿ R is a right invariant ring (e)=e for every central idempotent eA. The ring R/x ² R is right distributive R/x ⁿ R is right distributive for every positive integer n A is right or left Rickartian and right distributive, (e)=e for every central idempotent eA and the (a) is invertible in A for every regular aA. If A is a ring which is a finitely generated module over its center, then A[x] is a right Bezout ring A[x]/x ² A[x] is a right Bezout ring A is a regular ring.     

Extension of a class of fibered loops to kinematic spaces

The notion of the split extension of a commutative kinematic space is extended to the case of a weak K-loop with an incidence fibration (F, +, ). Theorem 1 states conditions under wich the quasi-direct productG F⁺ _Q with Aut(F, +) can be turned in a fibered incidence group (G, , o) such that (F, +, ) becomes embeddable inG, and Theorem 2 the additional assumption such that (G, , o) is even a kinematic space. In section 4, Theorem 3 shows that there are suitable examples of proper K-loops with an incidence fibration (derived from hyperbolic planes) on which one can apply Theorem 2.Dedicated to Erich Ellers on the occasion of his 70th birthdayResearch supported by M.U.R.S.T. 40% and by C.N.R. (G.N.S.A.G.A.)     

K0 of multiplier algebras of C*-algebras with real rank zero

For a large class of -unital C ^*-algebras A with real rank zero and stable rank one, the structure of the Grothendieck group k ₀ of the multiplier algebra (A) is investigated. The ordered group K ₀( (A)) is shown to be an unperforated Riesz group, and its additive structure is completely determined, as is — in important cases — its order structure. These structures, and the attendant consequences for the ideal structure of (A), are richer than previously anticipated. In particular, it is shown that the corona algebra (A)/A can have very large stably finite quotient algebras. For example, there exist simple, separable, approximately finite-dimensional C ^*-algebras A such that the maximal stably finite quotient algebra of (A)/A has uncountably many maximal ideals modulo which a W ^*-factor of Type II₁ results. The analysis of the additive structure of K ₀( (A)) yields as a byproduct that if A is a -unital approximately finite-dimensional C ^*-algebra without nonzero unital quotient algebras, then all quasitraces on (A) are traces.This research was partially supported by a grant from the National Science Foundation.     

Diameter of sets and measure of sumsets

For bounded sets A, B of reals we show that wherea=(A),b=(B) andD is the diameter ofB. For large values ofa this yields (A+B)a+D.Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.     

On the Approximability by Finite p-Groups of Free Products of Groups with Normal Amalgamation

A sufficient condition for the residual p-finiteness (approximability by the class of finite p-groups) of a free product G = (A * B; H) of groups A and B with a normal amalgamated subgroup H is obtained. This condition is used to prove that if A and B are extensions of residually -groups by -groups, where stands for the class of finitely generated torsion-free nilpotent groups, and if H is a normal p′-isolated polycyclic subgroup, then the group G is residually p-finite (i.e., residually -group), provided the quotient group G/H ^p H′ is residually p-finite.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 125–131.Original Russian Text Copyright © 2005 by E. V. Sokolov.     

On the lowest eigenvalue of the Laplacian for the intersection of two domains

IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for .There are two corollaries of this theorem: (i) A lower bound for sup _x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p ( _n ).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15     

On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.     

Le théorème limite central pour des sommes de Riesz-Raikov

Summary Let be an algebraic number greater than 1 andf a real 1-periodic function; ifF _N denotes the random variable defined on [0, 1] byF _N (t) , it is proved here that under sufficiently broad assumptions onf: 1) the sequence converges to a finite ²(0); 2) if >0, the sequence {F _N } converges in law to . We give an explicit computation of with respect to and a characterisation of functions for which =0.(Our results are also valid for almost every real >1).     

Einige Bemerkungen zum Goldbach'schen Problem

By combinatorial means the authors show the existence of thin sub-sets of primes, useful for Goldbach decompositions. For example, there is a set of primes with , such that all butO(x(ln x)^–A) even integersnx can be written as .

     

Description of Simple Exceptional Sets in the Unit Ball

For z B ⁿ, the boundary of the unit ball in . If the exceptional set for f. In this note we give a tool for describing such sets. Moreover we prove that if Eis a G and F subset of the projective (n– 1)-dimensional space then there exists a holomorphic function fin the unit ball B ⁿso that E(f) = E.     

Integration by parts for a Lie group valued Brownian motion

LetG be a Lie group ofd×d matrices and be theLLie algebra ofG. We choose some Euclidean norm on , and an orthonormal basis (D ₁,...D _m ) relative to it. Let be the corresponding left invariant vector fields onG. In this paper we derive an integration by parts formula for aG-valued Brownian motion corresponding to the Laplacian .     

Hamiltonian cycles in 1-tough graphs

For a graphG, let ₃ = min{ _i=1 ³ d(u_i): {u₁, u₂, u₃} is an independent set ofG} and = min{ _i=1 ³ d(u_i) – is an independent set ofG}. In this paper, we shall prove the following result: LetG be a 1-tough graph withn vertices such that ₃ n and – 4. ThenG is hamiltonian. This generalizes a result of Fassbender [2], a result of Flandrin, Jung and Li [3] and a result of Jung [5].Supported in part by das promotionsstipendium nach dem NaFöG and the Post-Doctoral Foundation of China.