On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

On the root closedness of continuous function algebras

For a compact Hausdorff space X, C(X) denotes the algebra of all complex-valued continuous functions on X. For a positive integer n, we say that C(X) is n-th root closed if, for each f∈C(X), there exists g∈C(X) such that f=gn. It is shown that, for each integer m?2, there exists a compact Hausdorff space Xm such that C(Xm) is m-th root closed, but not n-th root closed for each integer n relatively prime to m. This answers a question posed by Countryman Jr. [R.S. Countryman Jr., On the characterization of compact Hausdorff X for which C(X) is algebraically closed, Pacific J. Math. 20 (1967) 433-438] et al.     

Zero-separating algebras of continuous functions

Let A be a lattice-ordered algebra endowed with a topology compatible with the structure of algebra. We provide internal conditions for A to be isomorphic as lattice-ordered algebras and homeomorphic to Ck(X), the lattice-ordered algebra C(X) of real continuous functions on a completely regular and Hausdorff topological space X, endowed with the topology of uniform convergence on compact sets. As a previous step, we determine this topology among the locally m-convex topologies on C(X) with the property that each order closed interval is bounded.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

A remark on weakly convex continuous mappings in topological linear spaces

Let C be a compact convex subset of a Hausdorff topological linear space and T:C→C a continuous mapping. We characterize those mappings T for which T(C) is convexly totally bounded.     

Weakly compact operators and strict topologies

Let X   be a completely regular Hausdorff space and _Cb(X)$C_b(X)$ be the Banach space of all real-valued bounded continuous functions on X, endowed with the uniform norm. It is shown that every weakly compact operator T   from _Cb(X)$C_b(X)$ to a quasicomplete locally convex Hausdorff space E   can be uniquely decomposed as T=_T1+_T2+_T3+_T4$T=T_1+T_2+T_3+T_4$, where _Tk:_Cb(X)→E$T_k:C_b(X)\to E$(k=1,2,3,4)$(k=1,2,3,4)$ are weakly compact operators, and _T1$T_1$ is tight, _T2$T_2$ is purely τ  -additive, _T3$T_3$ is purely σ  -additive and _T4$T_4$ is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.     

Spaces of maps into topological group with the Whitney topology

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps endowed with the Whitney (graph) topology and by Cc(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l₂-manifold. In this article we show that if X is non-compact and not end-discrete then Cc(X,G) is an (R^∞×l₂)-manifold, and moreover the pair (C(X,G),Cc(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l₂.     

Unbounded Disjointness Preserving Linear Functionals

Let X be a locally compact Hausdorff space and C ₀(X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C ₀(X). They arise from prime ideals of C ₀(X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on C ₀(X) for arbitrary X. We also make some remarks where C ₀(X) is replaced by a non-commutative C*-algebra.     

Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Let C(X,T) be the group of continuous functions of a compact Hausdorff space X to the unit circle of the complex plane T with the pointwise multiplication as the composition law. We investigate how the structure of C(X,T) determines the topology of X. In particular, which group isomorphisms H between the groups C(X,T) and C(Y,T) imply the existence of a continuous map h of Y into X such that H is canonically represented by h. Among other results, it is proved that C(X,T) determines X module a biseparating group isomorphism and, when X is first countable, the automatic continuity and representation as Banach-Stone maps for biseparating group isomorphisms is also obtained.     

Surjective linear maps preserving local spectra

Let B(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize linear surjective and continuous maps on B(X) preserving different local spectral quantities at a nonzero fixed vector.     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

Complex function algebras and removable spaces

The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.     

Fragmentability of groups and metric-valued function spaces

Let (X,τ) be a topological space and let ρ be a metric defined on X. We shall say that (X,τ) is fragmented by ρ if whenever ε>0 and A is a nonempty subset of X there is a τ-open set U such that U∩A≠∅ and ρ−diam(U∩A)<ε. In this paper we consider the notion of fragmentability, and its generalisation σ-fragmentability, in the setting of topological groups and metric-valued function spaces. We show that in the presence of Baireness fragmentability of a topological group is very close to metrizability of that group. We also show that for a compact Hausdorff space X, σ-fragmentability of (C(X),‖⋅_‖∞) implies that the space Cp(X;M) of all continuous functions from X into a metric space M, endowed with the topology of pointwise convergence on X, is fragmented by a metric whose topology is at least as strong as the uniform topology on C(X;M). The primary tool used is that of topological games.     

Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.     

Spectral synthesis and other results in some topological algebras of vector-valued functions

Using results from theory of bundles of topological vector spaces, we prove some spectral synthesis and other results in certain topological algebras of vector-valued functions. In particular, we extend and generalize results of J.W. Kitchen and the second-named author (1996). We obtain as a corollary a recent (2008) result of Arizmendi-Peimbert/Carillo-Hoyo/García on the spectral synthesis property in (C _b (X),β), the space of bounded continuous functions on the completely regular Hausdorff space X under the strict topology β which arises from weighting by the non-negative upper semicontinuous functions on X which disappear at infinity.     

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.     

Local Uniform Strong Proximinality of the Space of Continuous Vector-Valued Functions

Let X be a Banach space, S be a compact Hausdorff space and Y be a U-proximinal subspace of X. We prove that C(S,Y) is locally uniformly strongly proximinal in C(S,X) and the corresponding metric projection map is Hausdorff metric continuous.