arXiv subjects

Shaymaa Al-shakarchi

Publications and source records attributed to Shaymaa Al-shakarchi.

4 recordsLinked to original sources

arXiv2021

Isomorphisms of $BV(σ)$ spaces

In this paper we investigate the relationship between the properties of a compact set $σ\subseteq \mathbb{C}$ and the structure of the space $BV(σ)$ of functions of bounded variation (in the sense of Ashton and Doust) defined on $σ$. For the subalgebras of absolutely continuous functions on $σ$, it is known that for certain classes of compact sets one obtains a Gelfand--Kolmogorov type result: the function spaces $AC(σ_1)$ and $AC(σ_2)$ are isomorphic if and only if the domain sets $σ_1$ and $σ_2$ are homeomorphic. Our main theorem is that in this case the isomorphism must extend to an isomorphism of the $BV(σ)$ spaces. An application is given to the spectral theory of $AC(σ)$ operators.

math.FA↗

arXiv2020

$AC(σ)$ spaces for polygonally inscribed curves

For certain families of compact subsets of the plane, the isomorphism class of the algebra of absolutely continuous functions on a set is completely determined by the homeomorphism class of the set. This is analogous to the Gelfand--Kolmogorov theorem for $C(K)$ spaces. In this paper we define a family of compact sets comprising finite unions of convex curves and show that this family has the `Gelfand--Kolmogorov' property.

math.FA↗

arXiv2020

Isomorphisms of $AC(σ)$ spaces for linear graphs

We show that among compact subsets of the plane which are drawings of linear graphs, two sets $σ$ and $τ$ are homeomorphic if and only if the corresponding spaces of absolutely continuous functions (in the sense of Ashton and Doust) are isomorphic as Banach algebras. This gives an analogue for this class of sets to the well-known result of Gelfand and Kolmogorov for $C(Ω)$ spaces.

math.FA↗

arXiv2017

Isomorphisms of $AC(σ)$ spaces for countable sets

It is known that the classical Banach--Stone theorem does not extend to the class of $AC(σ)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $σ$ is restricted to the set of compact polygons, then all the corresponding $AC(σ)$ spaces are isomorphic. In this paper we examine the case where $σ$ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic.

math.FA↗