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A connection between multiplication in C(X) and the dimension of X

Andrzej Komisarski (2006)

Fundamenta Mathematicae

Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.

A minimal regular ring extension of C(X)

M. Henriksen, R. Raphael, R. G. Woods (2002)

Fundamenta Mathematicae

Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring C ( X , τ δ ) of continuous real-valued functions on the space ( X , τ δ ) , where τ δ is the smallest Tikhonov topology on X for which τ τ δ and C ( X , τ δ ) is von Neumann regular. The compact and metric spaces for which G ( X ) = C ( X , τ δ ) are characterized. Necessary, and different sufficient, conditions...

A remark on supra-additive and supra-multiplicative operators on C ( X )

Zafer Ercan (2007)

Mathematica Bohemica

M. Radulescu proved the following result: Let X be a compact Hausdorff topological space and π C ( X ) C ( X ) a supra-additive and supra-multiplicative operator. Then π is linear and multiplicative. We generalize this result to arbitrary topological spaces.

Algebra of multipliers on the space of real analytic functions of one variable

Paweł Domański, Michael Langenbruch (2012)

Studia Mathematica

We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally m-convex. In particular, we show that Taylor multiplier sequences cease to be so after most permutations.

Algebrability of the set of non-convergent Fourier series

Richard M. Aron, David Pérez-García, Juan B. Seoane-Sepúlveda (2006)

Studia Mathematica

We show that, given a set E ⊂ 𝕋 of measure zero, the set of continuous functions whose Fourier series expansion is divergent at any point t ∈ E is dense-algebrable, i.e. there exists an infinite-dimensional, infinitely generated dense subalgebra of 𝓒(𝕋) every non-zero element of which has a Fourier series expansion divergent in E.

Algebras of real analytic functions: Homomorphisms and bounding sets

Peter Biström, Jesús Jaramillo, Mikael Lindström (1995)

Studia Mathematica

This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in l of the corresponding functions in A ( l ) . These results are achieved by studying the homomorphisms on the...

Algebras whose groups of units are Lie groups

Helge Glöckner (2002)

Studia Mathematica

Let A be a locally convex, unital topological algebra whose group of units A × is open and such that inversion ι : A × A × is continuous. Then inversion is analytic, and thus A × is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then A × has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group A × is an analytic Lie group without...

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