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.     

A representation theorem revisited

In 1962, the author proved that each reduced archimedean f-ring can be represented as an f-ring of continuous extended real-valued function on a locally compact space. The existence of this representation has proved to be quite useful; however, the proof so obscured the definition of the representing functions that deeper applications have remained out of reach. In this paper, we give a new proof of this result; one in which the derivation of the representing functions is more readily accessible. This accessibility is exploited to prove that f-ring homomorphisms on reduced archimedean f-rings are induced by continuous maps between subspaces of their representing spaces, and this leads to further insights into the structure of such homomorphisms. Presented by M. Henriksen. Received October 11, 2005; accepted in final form February 7, 2006. Many thanks to Tony Hager, who nagged me about this until I finally did something.     

A functional limit theorem for Erdös and Rényi's law of large numbers

Summary We use the theory of large deviations on function spaces to extend Erdös and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths {W _{N, n} } defined by where , {X _i : _i 1} is an iid sequence of random variables, andh(N)=[clogN] is relatively compact; the limit set is given by the set [xI ^*(x)1/c] whereI ^*(x) = ₀ ¹ I(x(t))dt andI is Cramér's rate function.     

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     

A New Characterization of the Continuous Functions on a Locale

Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c³A, the ``closed under countable composition hull of A,' as C(Y_lA), where Y_lA is the Yosida locale of A, and as the largest essential reflection of A.     

Some new results on functions in C(X) having their support on ideals of closed sets

Abstract

For any ideal of closed sets in X, let be the family of those functions in C(X) whose support lie on . Further let contain precisely those functions f in C(X) for which for each ? > 0, {x ∈ X: |f (x)| ≥ ?} is a member of . Let stand for the set of all those points p in βX at which the stone extension f^? for each f in is real valued. We show that each realcompact space lying between X and βX is of the form if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally- or almost locally-, becomes a space of the same form. We further show that is a free ideal (essential ideal) of C(X) if and only if is a free ideal (essential ideal) of when and only when X is locally- (almost locally-). We address the problem, when does or become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice ≡ the ideal of compact sets in X. Finally we observe that the ideals of the form of C(X) are no other than the z^?-ideals of C(X).     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

Inverse limits of arcs and of simple closed curves with confluent bonding mappings

It is proved that Knaster's type continua and solenoids can be considered as inverse limits of arcs and of circles with confluent bonding mappings. Several other classes of bonding mappings, which are relative to confluent ones, also are discussed.     

Gated sets in metric spaces

The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Extensions for Sobolev mappings between manifolds

We consider two compact Riemannian manifoldsM andN, such thatM has a boundary (but notN). We address the extension problem in the Sobolev class, namely, we investigate the question: foru W ^1–1/p,pM,N is there a mapV inW ^1/p(M,N) such thatV=u on M. Various results are given, and an emphasis is put on the special (simple) caseN=S ¹.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Set convergences: A survey and a classification

We endow families of closed subsets of a metric space with uniformities defined by semi-metrics. Such a structure is determined bytypes, which are semi-metrics defined by canonical functions on the metric space, andclasses, which are families of closed subsets.We obtain a classification of convergences on hyperspace by identifying a convergence to a pair (types, classes).The research of this author was done when he was visiting U.F.R.-M.I.M. of University of Provence in 1992.     

On the epicomplete monoreflection of an archimedean lattice-ordered group

In a category C an object it G is epicomplete if the only epic monics out of G are isomorphisms, epic or monic meant in the categorical sense of right or left cancellable. For each of the categories Arch: archimedean ?-groups with ?-homomorphisms, and its companion category W: Arch-objects with distinguished weak unit and unit-preserving ?-homomorphisms, (and for the corresponding categories of vector lattices) epicompleteness has been characterized as divisible and conditionally and laterally σ-complete, and it has been shown to be monoreflective. Denote the reflecting functors by β and β ^W , respectively. What are they? For W the Yosida representation has been used to realize β ^W A as a certain quotient of B (Y A), the Baire functions on the Yosida space of A. For Arch, very little has been known. Here we give a general representation theorem, Theorem A, for β G as a certain subdirect product of W-epicomplete objects derived from G. That result, some W-theory, and the relation between epicity and relative uniform density are then employed to show Theorem B: β C _K (Y)=B _L (Y), where C _K (Y)is the ?-group of continuous functions on Y with compact support and B _L (Y) is the ?-group of Baire functions on Y having Lindelöf cozero sets.     

Sacks forcing and the total failure of Martin's Axiom

Using side-by-side Sacks forcing, it is proved relatively consistent that the continuum is large and Martin's Axiom fails totally, that is, every c.c.c. space is the union of ?₁ nowhere dense sets (equivelently, if P is a nontrivial partial ordering with the countable chain condition, then there are ?₁ dense sets in P such that no filter in P meets them all).     

Characterizations of spaces by embeddings in βX

In this paper we obtain characterizations of metrizable spaces, paracompact M-spaces, Moore spaces and semimetrizable spaces in terms of the way those spaces are embedded in their Stone-?ech compactification. In addition, we give an internal characterization of paracompact M-spaces which we use in the proof of the embedding characterization.     

Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag _n ¹ , i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P ^N be the map associated to the line bundleK _S +[C] and letD be a general divisor of the given linear systemg _n ¹ . LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.     

Stable base loci, movable curves, and small modifications, for toric varieties

We show that the dual of the cone of divisors on a complete -factorial toric variety X whose stable base loci have dimension less than k is generated by curves on small modifications of X that move in families sweeping out the birational transforms of k-dimensional subvarieties of X. We give an example showing that it does not suffice to consider curves on X itself. Supported by a Graduate Research Fellowship from the NSF     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.