E *-Stable Theories

S. Shelah proved that stability of a theory is equivalent to definability of every complete type of that theory. T. Mustafin introduced the concept of being T ^*-stable, generalizing the notion of being stable. However, T ^*-stability does not necessitate definability of types. The key result of the present article is proving the definability of types for E ^*-stable theories. This concept differs from that of being T ^*-stable by adding the condition of being continuous. As a consequence we arrive at the definability of types over any P-sets in P-stable theories, which previously was established by T. Nurmagambetov and B. Poizat for types over P-models.     

First-order differentiability of the flow of a system withL p controls

This paper considers the study of the regularity of the flow of a nonautonomous nonlinear control process when the set of control maps is endowed with theL ^p -topology. Roughly speaking, it is proved that, if the norm of the mapf(t, x, u) defining the process together with its first derivatives goes to infinity, with the norm ofu not faster thanu ^p ,p>1, then the flow isC ¹ in theL ^p -topology. This property implies that, if the control maps are bounded, then the flow is differentiable in anyL ^p ,p>1. Moreover, it is proved that the only systems for which the flow is differentiable inL ¹ are the affine ones.This research was supported by a grant from Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica, Rome, Italy.     

F*-Rings Are O*

O ^*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O ^* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F ^*-rings. We show that the two classes of rings coincide.     

A New Construction of Central Relative (p a , p a , p a , 1)-Difference Sets

Semiregular relative difference sets (RDS) in a finite group E which avoid a central subgroup C are equivalent to orthogonal cocycles. For example, every abelian semiregular RDS must arise from a symmetric orthogonal cocycle, and vice versa. Here, we introduce a new construction for central (p ^a , p ^a , p ^a , 1)-RDS which derives from a novel type of orthogonal cocycle, an LP cocycle, defined in terms of a linearised permutation (LP) polynomial and multiplication in a finite presemifield. The construction yields many new non-abelian (p ^a , p ^a , p ^a , 1)-RDS. We show that the subset of the LP cocycles defined by the identity LP polynomial and multiplication in a commutative semifield determines the known abelian (p ^a , p ^a , p ^a , 1)-RDS, and give a second new construction using presemifields.We use this cohomological approach to identify equivalence classes of central (p ^a , p ^a , p ^a , 1)-RDS with elementary abelian C and E/C. We show that for p = 2, a 3 and p = 3, a 2, every central (p ^a , p ^a , p ^a , 1)-RDS is equivalent to one arising from an LP cocycle, and list them all by equivalence class. For p = 2, a = 4, we list the 32 distinct equivalence classes which arise from field multiplication. We prove that, for any p, there are at least a equivalence classes of central (p ^a , p ^a , p ^a , 1)-RDS, of which one is abelian and a – 1 are non-abelian.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

Spectral invariance of smooth crossed products, and rapid decay locally compact groups

We show that all rapid-decay locally compact groups are unimodular and that the set of rapid-decay functions on a locally compact rapidly decaying group forms a dense and spectral invariant Fréchet *-subalgebra of the reduced group C ^*-algebra. In general, the set of rapid-decay functions on a locally compact strongly rapid-decay group with values in a commutative C ^*-algebra forms a dense and spectral invariant Fréchet *-subalgebra of the twisted crossed product C ^*-algebra. The spectral invariance property implies that the K-theories of both algebras are naturally isomorphic under inclusion.This project is supported in part by the National Science Foundation Grant #DMS 92-04005.     

Harmonic Functions,Entropy, and a Characterization of the Hyperbolic Space

Let (M ⁿ ,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ ₀ of its universal covering satisfies λ ₀≤(n−1)²/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. The author was partially supported by NSF Grant 0505645.     

Operating characteristics of MX/G/1 queue with N-policy

We consider aM ^X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM ^X/G/1 queueing system withoutN-policy and the other one has the probability generating function _j=0 ^N=1 _j z ^j/ _j=0 ^N=1 _j , in which _j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.     

Weak-type Endpoint Estimates for Multilinear Singular Integral Operators

In this paper, the authors discuss a class of multilinear singular integrals and obtain that the operators are bounded from H^1(R^n) to weak L^1(R^n). Using this result, we can directly prove a main theorem in [5].     

Lp-Lq Estimates for higher order perturbed hyperbolic equation

In this paper L^p-L^q estimates for the solution u(x,t) to the following perturbed high-er order hyperbolic equation are considered, (ρπ--a△)(ρπ--b△)u V(x)u=O, x∈R^n,n≥6, ρ1eu(x,O) = O, ρ^3eu(x,O) = f(x), (j = O,1,2).We assume that the otential V(x) and the initial data f(x) are compactly supported, andV(x) is sufficiently small, then the solution u (x,t) of the above problem satisfies the same L^p-L^q estimates as that of the unperturbed problem.     

Fundamental groups of manifolds with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁,W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹ → M ⁿ . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.      

Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L^p-functional calculi, or may be of holomorphic L^p-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic L^p-type” we mean that every L^p-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L²-spectrum of L. This can only arise if the group algebra L¹(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L¹(G) [37], also suffices for L to be of holomorphic L¹-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic L^p-type for every p≠2. Besides the non-symmetry of L¹(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L¹-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

On classifying monotone complete algebras of operators

We give a classification of “small” monotone complete C ^*-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 ^c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C ^*-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of ℓ^∞. Some examples and applications are given.      

Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C ¹ or C ²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C ¹ or C ²-curves. Surprisingly, what is true for C ¹-curves fails to be true in part for C ²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.     

直线上子集的H~s-拓扑及其应用

本文利用s-维Hausdorff测度给出了直线上一个子集E上的H~s拓扑和H~s-连通度的定义.讨论了它们的性质及其应用,解决了紧的s-集在欧氏拓扑下往往不连通的问题.     

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed&gt;1 and {0} if d&gt;1.Let Bcd be a cd-homogeneous C^*-algebra over S^2&#215;T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2&#215;Z^m-2) in Bcd &#215;C^*(Z^m-3) by a totally skew multiplier ρ on T^2&#215;Z^m-2。It is shown that Sρ^cd&#215;Mρ∞ is isomorphic to C(S^2)&#215;C^*(T^2&#215;Z^m-2,ρ）&#215; Mcd(C)&#215;Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.