Superatomic Boolean algebras constructed from strongly unbounded functions

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ^{++ +} ≤ λ, κ^<κ = κ and 2^κ = κ⁺, and η is an ordinal with κ⁺ ≤ η < κ⁺⁺ and cf(η) = κ⁺. Then, in some cardinal‐preserving generic extension there is a superatomic Boolean algebra $\mathcal BUsing Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ^{++ +} ≤ λ, κ^<κ = κ and 2^κ = κ⁺, and η is an ordinal with κ⁺ ≤ η < κ⁺⁺ and cf(η) = κ⁺. Then, in some cardinal‐preserving generic extension there is a superatomic Boolean algebra $\mathcal B$ such that $\mathrm{ht}(\mathcal B) = \eta + 1$, $\mathrm{wd}_{\alpha }(\mathcal B) = \kappa$ for every α < η and $\mathrm{wd}_{\eta }(\mathcal B) = \lambda$(i.e., there is a locally compact scattered space with cardinal sequence 〈κ〉_η^??〈λ〉). Especially, $\langle {\omega }\rangle _{{\omega }_1}{}^{\smallfrown } \langle {\omega }_3\rangle$ and $\langle {\omega }_1\rangle _{{\omega }_2}{}^{\smallfrown } \langle {\omega }_4\rangle$ can be cardinal sequences of superatomic Boolean algebras.     

Weighted algebras of continuous functions

We give conditions under which the weighted spaces CV(X) and CV₀ (X) are locally convex algebras of a certain type and proceed to a systematic study of these algebras. We show, in particular, that the spectrum of CV₀ (X) is always homeomorphic to X and we detemine the spectrum of CV(X).     

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.     

Homomorphisms of weighted algebras of continuous functions

Summary A characterization is given of all continuous algebra homomorphisms H from the weighted space of continuous functions CV(X, E) into ℂ, where X is a completely regular Hausdorff space, E a locally convex algebra and V a family of weight functions that vanish at infinity. These homomorphisms are represented in the form H(f) = h(f(x)) with x in X and h a continuous algebra homomorphism on E. We then consider the space of homomorphisms of CV(X, E) and give some counterexamples to possible further generalizations. Entrata in Redazione il 18 febbraio 1977. ? Aspirant ? of the Belgian ? Nationaal Fonds voor Wetenschappelijk Onderzoek ?.     

Polynomial approximations of continuous functions with unbounded sign-sensitive weight

The conditions of arbitrarily exact polynomial approximation of functions continuous on a closed interval with respect to an unbounded sign-sensitive weight are obtained by using divided differences with multiple nodes. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Quotient spaces determined by algebras of continuous functions

We prove that if X is a locally compact σ-compact space, then on its quotient, γ(X) say, determined by the algebra of all real valued bounded continuous functions on X, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if X is second countable locally compact, then γ(X) is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in [1] and [7] where the primitive ideal space of a C*-algebra was considered.     

Harmonic generalized functions in generalized function algebras

We investigate basic properties of harmonic generalized functions within the framework of J. F. Colombeau??s theory of generalized functions. In particular, we present various theorems concerning the Maximum principle, Liouville??s theorem, singularities and Poisson formula.     

Lie algebras of unbounded derivations

In this paper some new results on analytic domination of operators and on integrability of Lie algebras of operators are proved and then our methods are applied to the study of Lie algebras of unbounded derivations in $\text{C}{}^1$ algebras.     

On strict homological dimensions of algebras of continuous functions

We prove that, for each nonnegative integer n and n = ∞, there exists a compact topological space Ω such that the strict global dimension and the strict bidimension of the Banach algebra C(Ω) of all continuous functions on Ω are equal to n. We also obtain several “additivity formulas” for the strict homological dimensions of strict Banach algebras.     

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max(A) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.     

Calculating quotient algebras of generic embeddings

Many consistency results in set theory involve forcing over a universe V ₀ that contains a large cardinal to get a model V ₁. The original large cardinal embedding is then extended generically using a further forcing by a partial ordering ?. Determining the properties of ? is often the crux of the consistency result. Standard techniques can usually be used to reduce to the case where ? is of the form P(Z)/J for appropriately chosen Z and countably complete ideal J. This paper proves a general algebraic Duality Theorem that exactly characterizes the Boolean algebra P(Z)/J. The Duality Theorem is general enough that it applies even if the original embedding in V ₀ was itself generic. Thus it has as corollaries the theorems of Kakuda, Baumgartner, Laver and others about preservation properties of precipitous and saturated ideals. A corollary is drawn showing that precipitous ideals are indestructible under small proper forcing.     

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.