Porosity and diametrically maximal sets in C(K)

This paper is motivated by recent attempts to investigate classical notions from finite-dimensional convex geometry in spaces of continuous functions. Let be the family of all closed, convex and bounded subsets of C(K) endowed with the Hausdorff metric. A completion of is a diametrically maximal set satisfying A ⊂ D and diam A = diam D. Using properties of semicontinuous functions and an earlier result by Papini, Phelps and the author [12], we characterize the family γ(A) of all possible completions of . We give also a formula which calculates diam γ(A) and prove finally that, if K is a Hausdorff compact space with card K > 1, then the family of those elements of having a unique completion is uniformly very porous in with a constant of lower porosity greater than or equal to 1/3.     

Essential ideals inC(X)

It is shown thatX is finite if and only ifC(X) has a finite Goldie dimension. More generally we observe that the Goldie dimension ofC(X) is equal to the Souslin number ofX. Essential ideals inC(X) are characterized via their corresponding z-filters and a topological criterion is given for recognizing essential ideals inC(X). It is proved that the Fréchet z-filter (cofinite z-filter) is the intersection of essential z-filters. The intersection of idealsO _x wherex runs through nonisolated points inX is the socle ofC(X) if and only if every open set containing all nonisolated points is cofinite. Finally it is shown that if every essential ideal inC(X) is a z-ideal thenX is a P-space.     

Remarks on intermediate C-rings of C(X)

Let X be a Tychono? space and A(X) be a subring of C(X) containing C^?(X). We introduce the notion of -ideal in A(X). It is observed that the class of -ideals contains the class of z_A-ideals and is contained in the class of z-ideals of A(X). These containments may be proper. It turns out that coincidence of z-ideals of A(X) with -ideals characterizes intermediate C-rings of C(X).     

Constant curved minimal 2-spheres in G(2,4)

This paper gives a complete classification for minimal 2-spheres with constant Gaussian curvature immersed in the complex Grassmann manifold G(2,4). Received: 14 May 1998 / Revised version: 12 October 1998     

Some remarks on category of the real line

We find a characterization of the covering number , of the real line in terms of trees. We also show that the cofinality of is greater than or equal to for every where ( is the additivity number of the ideal of all Lebesgue measure zero sets) is the least cardinal number k for which the statement: fails. Received: 19 October 1994 / Revised version: 12 December 1996     

Lattice and Algebra Homomorphisms on C(X) in Zermelo-Fraenkel Set Theory

Abstract

Let C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice homo- morphisms, and point evaluations.     

Homomorphisms on Function Lattices

In this paper we study real lattice homomorphisms on a unital vector lattice , where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.     

Isomorphisms of algebras of smooth functions revisited

It is proved that isomorphisms between algebras of smooth functions on Hausdorff smooth manifolds are implemented by diffeomorphisms. It is not required that manifolds are connected nor second countable nor paracompact. This solves a problem stated by A. Weinstein. Some related results are discussed as well.Received: 3 November 2004     

On Subspaces and Quotients of Banach Spaces C(K, X)

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces. Received 8 November 2000; in revised form 7 December 2001     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999     

Sampling measures for Bergman spaces on the unit disk

We provide a characterization of the sampling measures for the Bergman spaces. These are the positive measures on the unit disk for which there exists a constant such that These are the continuous analogues of the sets of sampling characterized by K. Seip [13,14] and A. Schuster [12]. Our characterization is in terms of weak* limits of the Moebius transformations of the measure , and mimics the notion for sequences that sampling means being uniformly far from zero sets. Received: 26 October 1998 / in revised form: 25 Juni 1999     

Quotient rings of algebras of functions and operators

Abstract. Various quotient rings of rings B of Banach algebra A-valued continuous functions on a completely regular Hausdorff Space X are constructed in terms of continuous functions defined on dense open subsets of X taking values in the maximal quotient ring of the Banach algebra A. This extends the results proved by N. J. Fine, L., Gillman and J. Lambek (1965) for the case of A, the field of real numbers. The pattern is similar and utilizes as well as generalises the results proved for algebras of multipliers of B by C. A. Akemann, G. K. Pedersen and J. Tomiyama (1973). The techniques combine those from algebra, analysis and topology. The details of the cases when A is the normed division algebra of real quaternions or the operator algebra B(H) of a Hilbert space H are given to illustrate our results. Received February 1, 1999; in final form June 18, 1999 / Published online May 8, 2000     

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001     

A new factorization technique of the matrix mask of univariate refinable functions

Summary. A univariate compactly supported refinable function can always be written as the convolution product , with the B-spline of order k,f a compactly supported distribution, and k the approximation orders provided by the underlying shift-invariant space . Factorizations of univariate refinable vectors were also studied and utilized in the literature. One of the by-products of this article is a rigorous analysis of that factorization notion, including, possibly, the first precise definition of that process. The main goal of this article is the introduction of a special factorization algorithm of refinable vectors that generalizes the scalar case as closely (and unexpectedly) as possible: the original vector is shown to be `almost' in the form , with F still compactly supported and refinable, andk the approximation order of . The algorithm guarantees F to retain the possible favorable properties of , such as the stability of the shifts of and/or the polynomiality of the mask symbol. At the same time, the theory and the algorithm are derived under relatively mild conditions and, in particular, apply to whose shifts are not stable, as well as to refinable vectors which are not compactly supported. The usefulness of this specific factorization for the study of the smoothness of FSI wavelets (known also as `multiwavelets' and `multiple wavelets') is explained. The analysis invokes in an essential way the theory of finitely generated shift-invariant (FSI) spaces, and, in particular, the tool of superfunction theory. Received June 10, 1998 / Revised version received June 14, 1999 / Published online August 2, 2000     

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

This article focuses on the study of the metric geometry of homogeneous spaces (the unitary group of a C*-algebra modulo the unitary group of a C*-subalgebra ) where the invariant Finsler metric in is induced by the quotient norm of Under the assumption that is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points there exists a minimal uniparametric group curve joining ρ₀ and ρ₁.     

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated. Author’s address: J. Li and F. Lu, Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China This research was supported by NNSFC (No. 10771154) and PNSFJ (NO. BK2007049).     

Complete minimal hypersurfaces in complex hyperbolic space

We construct new examples of embedded, complete, minimal hypersurfaces in complex hyperbolic space, including deformations of bisectors and some minimal foliations. Received: 20 March 2000 / Revised version: 21 July 2000