Topology Of Simple Convergence Research Articles

The Corona problem on a complete ultrametric algebraically closed field

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult m (A, ‖. ‖) be the subset of the ϕ ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult 1(A, ‖. ‖) be the subset of the ϕ ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult 1(A, ‖. ‖) is dense in Mult m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ℂ p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult 1(A, ‖. ‖), which suggest that Mult 1(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).

Subgroups of isometries of Urysohn–Katětov metric spaces of uncountable density

According to Katětov (1988) [15], for every infinite cardinal m satisfying mn⩽m for all n<m, there exists a unique m-homogeneous universal metric space Um of weight m. This object generalizes the classical Urysohn universal metric space U=Uℵ0. We show that for m uncountable, the isometry group Iso(Um) with the topology of simple convergence is not a universal group of weight m: for instance, it does not contain Iso(U) as a topological subgroup. More generally, every topological subgroup of Iso(Um) having density <m and possessing the bounded orbit property (OB) is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskijʼs (1990) [35] result about the group Iso(U) being a universal Polish group.

Universal spaces of subdifferentials of sublinear operators ranging in the cone of bounded lower semicontinuous functions

We study Frechet’s problem of the universal space for the subdifferentials ∂P of continuous sublinear operators P: V → BC(X)∼ which are defined on separable Banach spaces V and range in the cone BC(X)∼ of bounded lower semicontinuous functions on a normal topological space X. We prove that the space of linear compact operators Lc(l2, C(βX)) is universal in the topology of simple convergence. Here l2 is a separable Hilbert space, and βX is the Stone-Ĉech compactification of X. We show that the images of subdifferentials are also subdifferentials of sublinear operators.

The Ultrametric Corona Problem and spherically complete fields

AbstractLet K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the ‘open’ unit disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Multa(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Multa(A, ‖ · ‖) dense in Multm(A, ‖ · ‖)? Is it dense in Multm(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ‖ · ‖) in Multm(A, ‖ · ‖).

On Sequential Separability

A new criterion for the sequential separability of the space C(X) of continuous real-valued functions on X in the topology of simple convergence is obtained.

A weakly stable method for linear integral equations of the first kind

Studies a new method of a weak approximation for a stable solution of some first kind integral equations ∫ abK(x,y)u(x)dx=f(y) with continuous kernel. It is well known that the problem of resolution of these equations is ill‐posed in the sense of the methods that use the strong norms such as the one of uniform convergence. The method developed here is based on a weak formulation of the problem enabling us to weakly approach the solution and guarantee some weak stability with respect to the topology of simple convergence. An application to the inverse Laplace transform is treated at the end of this paper.

Le degré de Lindelöf est $l$-invariant

The Lindelof number $l(X)$ of a Tychonoff space $X$ is the smallest infinite cardinal $\tau$ such that any open cover of $X$ contains a subcover of cardinality less than or equal to $\tau$. The symbol $C_p(X)$ denotes the space of real-valued continuous functions on $X$ endowed with the topology of simple convergence. A well known fact is that if $C_p(X)$ and $C_p(Y)$ are isomorphic as topological rings, then $X$ and $Y$ are homeomorphic. The main resul of this paper shows that if $C_p(X)$ and $C_p(Y)$ are linearly homeomorphic, then $l(X)=l(Y)$.

Bornological spaces of non-Archimedean valued functions

Let C(X,E) be the space of all continuous functions from an ultraregular space X to a non-Archimedean locally convex space E. Necessary and/or sufficient conditions are given so that C(X,E), with the topology of uniform convergence on compact sets or with the topology of simple convergence, is bornological or c-ultrabornological.

On the topology of simple convergence in non-Archimedean function spaces

Let X be an ultraregular Hausdorff space, F a non-trivial complete nonArchimedean valued field and E a non-Archimedean F-locally convex space over F. Let C,(X, E) denote the space of all continuous E-valued functions on X equipped with the topology of simple convergence. In this paper, we study some of the properties of the space C,(X, E). We look at the problems: When is this space barrelled, quasi-barrelled or bornological? For the bornologicity of C,(X, E) we show that, when E is a non-Archimedean normed space, the space C,(X, E) is bornological iff X is N-replete. This is analogous to the result of Govaerts [4] for the space C,(X, F). We also characterize the strongly bounded subsets of the dual space of C,(X, E). In case E is a non-Archimedean normed space, it is shown that:

Disintegration with respect to L p-density functions and singular measures

Let A be a real or complex commutative ordered algebra with identity and involution. Let Г denote the set of positive multiplicative linear functionals ρ on A. Equip Г with the topology of simple convergence. For a fixed non-negative probability measure μ on Г the set ℒ p of linear functionals f on A which admit an integral representation of the form $$f(x) = \mathop \smallint \limits_r \rho (x)F(\rho )d\mu (\rho )$$ with F∈L p (μ) (1≦p≦τ) is biuniquely identified with L p (μ) via the map tf→F. The norm on ℒ p under which this map becomes an isometry is characterized and a formula for approximating F is derived. The linear functionals which admit representation of the form $$\mathop \smallint \limits_r \rho (x)dv(\rho )$$ with ν⊥μ are also characterized and appropriately normed. The theory is applied to solve abstract versions of trigonometric and n-dimensional moment problems as well as provide an alternate point of view to the theory of L p-spaces. New proofs of classical theorems are offered.

On the spaceC(X, E) with the topology of simple convergence

On the spaceC(X, E) with the topology of simple convergence

Une nouvelle définition des cônes biréticulés

It is proved that, if E is a vector lattice, the cone of positive linear forms on E, endowed with the topology of simple convergence on E is a bireticule cone. This result leads to a new definition of the bireticule cones, equivalent to the initial one, but much more handy ; it answers positively to an hypothesis made by G. Choquet.