On Separation of Points from Additive Subgroups of Banach Spaces by Continuous Characters and Positive Definite Functions

Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions: (ws) if , then x is weakly separated from G; (ps) if , then x is -separated from G; (wp) if Tx is -separated from T(G), then x is weakly separated from G. By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .      

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family X_z, , over the closed unit disc , of real analytic plane Jordan curves X_z. Ifj _e ^iθ ,e ^iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of fixingX _e ^iθ pointwise, we show that there is a unique holomorphic motion of extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X₀.     

Groupes Moyennables, Dimension Topologique Moyenne et sous-décalages

Let G be an infinite countable residually finite amenable group. In this paper we construct a continuous action of G on a compact metrisable space X such that the dynamical system (X, G) cannot be embedded in the G-shift on [0,1] ^G . This result generalizes a construction due to E. Lindenstrauss and B. Weiss (Mean topological dimension, Israel J. Math. 115 (2000), 1–24) for .     

On amenability of group algebras, I

We study amenability of algebras and modules (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring G is amenable for some (and therefore for any) field . Similarly, a G-set X is amenable if and only if its span X is amenable as a G-module for some (and therefore for any) field .     

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.     

A Remark on Characterizations of Projective Spaces among Singular Varieties

Let X be a projective variety of dimension n ≥ 2 with at worst log-terminal singularities and let be an ample vector bundle of rank r. By partially extending previous results due to Andreatta and Wiśniewski in the smooth case, we prove that if r = n then , while if r = n − 1 and X has only isolated singularities, then either or n = 2 and X is the quadric cone Q ₂. Received: April 20, 2006. Revised: April 5, 2007.     

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism , where B _n ¹ is a braid group of type B. The formality isomorphism depends algebraically on a series Ψ_KZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded (N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue (N, k) of (N, k); it is a graded Lie algebra with an action of , and we give a lower bound for the character of its space of generators.      

Simple immersions of wonderful varieties

Let G be a semisimple connected linear algebraic group over , and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample. Research supported by European Research Training Network LIEGRITS (MRTN-CT 2003- 505078), in contract with CNRS DR17, No 2.     

Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .      

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient , for certain external subspaces of the hyperfinite dimensional Banach space , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and are isometrically isomorphic as Banach algebras. Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic.     

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

Free Lie algebroids and the space of paths

We construct algebraic and algebro-geometric models for the spaces of unparametrized paths. This is done by considering a path as a holonomy functional on indeterminate connections. For a manifold X, we construct a Lie algebroid which serves as the tangent space to X (punctual paths) inside the space of all unparametrized paths. It serves as a natural receptacle of all “covariant derivatives of the curvature” for all bundles with connections on X. If X is an algebraic variety, we integrate to a formal groupoid which can be seen as the formal neighborhood of X inside the space of paths. We establish a relation between and the stable map spaces of Kontsevich.      

On meromorphic solutions of f 3 + g 3 = 1

We show that meromorphic solutions f, g of f ³ + g ³ = 1 in must be constant, if and have the same zeros (counting multiplicities). Received: 12 March 2007     

Some New Observations on Interpolation in the Spectral Unit Ball

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ω _n , n ≥ 2. We begin by showing that a known necessary condition for the existence of a -interpolant ( here being the unit disc in ), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem – one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ω _n , n ≥ 2. This work is supported in part by a grant from the UGC under DSA-SAP, Phase IV.     

Hyperbolic unit groups and quaternion algebras

We classify the quadratic extensions and the finite groups G for which the group ring [G] of G over the ring of integers of K has the property that the group of units of augmentation 1 is hyperbolic. We also construct units in the ℤ-order of the quaternion algebra , when it is a division algebra.     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

Boundedness, regularity and smoothness of universal Taylor series

Let Ω be an unbounded simply connected domain in satisfying some topological assumptions; for example let Ω be an open half-plane. We show that there exists a bounded holomorphic function on Ω which extends continuously on and is a universal Taylor series in Ω in the sense of Luh and Chui–Parnes with respect to any center. Our proof uses Arakeljan’s Approximation Theorem. Further we strengthen results of G. Costakis [2] concerning universal Taylor series with respect to one center in the sense of Luh and Chui–Parnes in the complement G of a compact connected set. We prove that such functions can be smooth on the boundary of G and be zero at ∞. If the universal approximation is also valid on ∂G, then the function can not be smooth on ∂G, but it may vanish at ∞. Our results are generic in natural Fréchet spaces of holomorphic functions. Received: 29 September 2005; revised: 21 February 2006     

<Emphasis Type="Italic">G</Emphasis>-linear sets and torsion points in definably compact groups

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G\X) < dim G for some definable then X contains a torsion point of G. Along the way we develop a general theory for the so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G. M. Otero was Partially supported by GEOR MTM2005-02568 and Grupos UCM 910444.