FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS

Abstract

This paper investigates torsion-free abelian groups A which are Q E-flat, i.e. for which Q A is flat as an Q E(A)-module. It is shown that a torsion-free A has this property iff Tor¹ (M, A) is torsion for all right E(A)-modules M. Furthermore, a torsion-free group of rank 4 is constructed which is Q E-flat but not quasi-isomorphic to an E-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which are Q E-flat.     

On existence subspaces inC(Q)

The property of a space to be an existence subspace is studied for subspacesE ofC(Q) such that eitherE orC(Q)/E is a Lindenstrauss space. For a Chebyshev subspaceL⊂C(Q)₁ an analytic representation of the nearest element in terms of the annihilatorL ^┴ is obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 726–737, May, 1999.     

Potential Analysis on Carnot Groups，Part II：Relationship between Hausdorff Measures and Capacities

Abstract In this paper, we establish the relationship between Hausdorff measures and Bessel capacities on any nilpotent stratified Lie group (i. e., Carnot group). In particular, as a special corollary of our much more general results, we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of . Given any set E ⊂ , B _α,p (E) = 0 implies ℋ ^{Q−αp+ ε}(E) = 0 for all ε > 0. On the other hand, ℋ ^Q−αp (E) < ∞ implies B _α,p (E) = 0. Consequently, given any set E ⊂ of Hausdorff dimension Q − d, where 0 < d < Q, B _α,p (E) = 0 holds if and only if αp ≤ d. A version of O. Frostman’s theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4). Research supported partly by the U. S. National Science Foundation Grant No. DMS99–70352     

Uniform approximation: The non-locally convex case

IfS is a compact Hausdorff space of finite covering dimension and (E, τ) is a real or complex topological vector space (not necessarily locally convex), we prove a Weierstrass-Stone theorem for subsets ofC(S;E), the space of all continuous functions fromS intoE, equipped with the topology of uniform convergence overS.     

Symetries isometriques applications a la dualite des espaces reticules

We show that ifE is a non-reflexive Banach lattice, there exists for everyn a dual of finite even order ofE which contins isometicallyl _n ^/l . We show that itE is a Banach lattice which is isometric to the dual of a Banach spaceX, then the order intervals are σ (E, X)-compact. We prove then that under various conditions, a Banach lattice which is a dual as a Banach space, is a dual as a Banach lattice. In particular, this is true when the predual ofE is unique.      

p-Lattice Summing Operators

In this paper we study the spaces ∞_p(E, X) of p-lattice summing operators from a Banach space E to a Banach lattice X. The main results characterize those E and X for which Δ_p(E, X) = II_p(E, X) and we show that ∞(E, X)=Δ₂(E, X) for an infinite dimensional Banach lattice X of finite cotype if and only if E is isomorphic to a Hilbert space.     

Spaces of Vector-Valued Integrable Functions and Localization of Bounded Subsets

We study the structure of bounded sets in the space L¹{E} of absolutely integrable Lusin-measurable functions with values in a locally convex space E. The main idea is to extend the notion of property (B) of Pietsch, defined within the context of vector-valued sequences, to spaces of vector-valued functions. We prove that this extension, that at first sight looks more restrictive, coincides with the original property (B) for quasicomplete spaces. Then we show that when dealing with a locally convex space, property (B) provides the link to prove the equivalence between Radon–Nikodym property (the existence of a density function for certain vector measures) and the integral representation of continuous linear operators T: L¹ → E, a fact well-known for Banach spaces. We also study the relationship between Radon–Nikodym property and the characterization of the dual of L¹{E} as the space L^∞{E′_b}.     

A Hilbert Space Approach to Effective Resistance Metric

A resistance network is a connected graph (G, c). The conductance function c _xy weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form E{\mathcal E} produces a Hilbert space structure (which we call the energy space H_E{{\mathcal H}_{\mathcal E}}) on the space of functions of finite energy. We use the reproducing kernel {v _x } constructed in a previous work to analyze the effective resistance R, which is a natural metric for such a network. It is known that when (G, c) supports nonconstant harmonic functions of finite energy, the effective resistance metric is not unique. The two most natural choices for R(x, y) are the “free resistance” R ^F , and the “wired resistance” R ^W . We define R ^F and R ^W in terms of the functions v _x (and certain projections of them). This provides a way to express R ^F and R ^W as norms of certain operators, and explain R ^F ≠  R ^W in terms of Neumann versus Dirichlet boundary conditions. We show that the metric space (G, R ^F ) embeds isometrically into H_E{{\mathcal H}_{\mathcal E}}, and the metric space (G, R ^W ) embeds isometrically into the closure of the space of finitely supported functions; a subspace of H_E{{\mathcal H}_{\mathcal E}}. Typically, R ^F and R ^W are computed as limits of restrictions to finite subnetworks. A third formulation R ^tr is given in terms of the trace of the Dirichlet form E{\mathcal E} to finite subnetworks, and is related to R ^F by a probabilistic argument.     

Representations of Hardy Algebras: Absolute Continuity,Intertwiners, and Superharmonic Operators

Suppose T₊(E){\mathcal{T}_{+}(E)} is the tensor algebra of a W*-correspondence E and H ^∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T₊(E){\mathcal{T}_{+}(E)} on a Hilbert space to ultra-weakly continuous completely contractive representations of H ^∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.     

Essential and inessential points of local nonconvexity

LetS be a closed connected subset of a Hausdorff linear topological space,Q the points of local nonconvexity ofS, E the essential members ofQ, N the inessential. IfS～Q is connected, then the following are true: Theorem 1.If Q ?Øis countable, then S is planar. Theorem 2.If Q is finite and nonempty, then cardE≧cardN+1. Theorem 3.If SυR ² and N is infinite, then E is infinite.     

The Order Properties of r-compact Operators on Banach Lattices

We present here that F（E,F）, the space of all r-compact operators from E into F, is a generalised sublattice of L^r（E, F） for arbitary Banach lattices E and F, and that the characterization of the regular norm on F（E, F） is order continuous. Some conditions for F（E, F） to be a KB-space or a band in .L（E, F） are also provided.     

Formal groups and zeta-functions of elliptic curves

The author shows that the isomorphism class of a formal group overZ/pZ (resp. overZ _p ) of finite height (resp. having reduction modp of finite height) is determined by its characteristic polynomial. It is then proved that the formal groups associated to a large class of Dirichlet series with integer coefficients are defined overZ.Finally, these results are used to extend a theorem of Honda (Osaka J. Math.5, 199–213 (1968), Theorem 5) to include the case of supersingular reduction at the primes 2 and 3. LetE be an elliptic curve defined overQ, andF(x, y) be a formal minimal model forE. LetG(x, y) be the formal group associated to the globalL-seriesL(E, s) ofE overQ. Honda's theorem now becomes:G(x, y) is defined over Z and is isomorphic over Z to F(x, y).     

Q-curves of degree 5 and jacobian surfaces of GL2-type

We construct a parametric family {E ^(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q , and the family {C(s,t,u)} of genus two curves over Q covering E ^{(+)(s,t,u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E ^{(+)(s,t,u) and its twist E ⁽⁻⁾(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509. Received: 23 September 1997 / Revised version: 26 May 1998     

Maximal Chains in Positive Subfamilies of P(ω)

A family P ì [w]^w{\mathcal P} \subset [\omega]^\omega is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/Fin. For example, if I ì P(w){\mathcal I} \subset P(\omega) is an ideal containing the ideal Fin of finite subsets of ω, then P(w) \IP(\omega) \setminus {\mathcal I} is a positive family and the set Dense(\mathbb Q)\mbox{Dense}({\mathbb Q}) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(\mathbb Q)P({\mathbb Q}). We prove that, for a positive family P{\mathcal P}, the order types of maximal chains in the complete lattice áP è{?}, ì ?\langle {\mathcal P} \cup \{\emptyset\}, \subset \rangle are exactly the order types of compact nowhere dense subsets of the real line having the minimum non-isolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0,1)_{\mathbb R}\mbox{Intalg}[0,1)_{{\mathbb R}} and the poset E(\mathbb Q)E({\mathbb Q}), where E(\mathbb Q)E({\mathbb Q}) is the set of elementary submodels of the rational line.     

Computing the nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (Δ_E,Δ_A) such that (E+Δ_E,A+Δ_A) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair (E,A) is dissipative Hamiltonian if A=(J−R)Q with skew‐symmetric J, positive semidefinite R, and an invertible Q such that Q^TE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.     

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i? _t u + Δu + |u|² u = 0 such that ‖xu(t)‖ _{L ²}  = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|² Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u ₀‖ _{L ²} ‖?u ₀‖ _{L ²}  > ‖Q‖ _{L ²} ‖?Q‖ _{L ²} , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t _n  → + ∞ such that ‖?u(t _n )‖ _{L ²}  → ∞. Similar statements hold for negative time.     

On the square root of a positive B( )-valued function

Let (Ω, , μ) be a measure space, a separable Banach space, and ^* the space of all bounded conjugate linear functionals on . Let f be a weak^* summable positive B( ^*)-valued function defined on Ω. The existence of a separable Hilbert space , a weakly measurable B( )-valued function Q satisfying the relation Q^*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺( ^*)-valued measures, the concepts of weak^*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Construction of markov process with markov resolvent onL P (E, m)

LetE be a Hausdorff topological space and let β(E) be its Borel σ-field. Letm be a σ-finite measure on (E, β(E)). A necessary and sufficient condition for a Markov resolvent onL ^P (E,m) to be associated with anm-tightm-special standard process (with state spaceE) is given. Furthermore some new examples which do not belong to the framework of Dirichlet space are also given. Project supported by the Youth Foundation of Chinese Academy of Sciences.     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

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Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

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Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno