Jointly separating maps between vector-valued function spaces

Year, pages: 2020, 707 - 718

Let $X$ and $Y$ be compact Hausdorff spaces, $E$ be a real or complex Banach space and $F$ be a real or complex locally convex topological vector space. In this paper we study a pair of linear operators $S,T: A(X,E) \lo C(Y,F)$ from a subspace $A(X,E)$ of $C(X,E)$ to $C(Y,F)$, which are jointly separating, in the sense that $Tf$ and $Sg$ have disjoint cozeros whenever $f$ and $g$ have disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pair $T: A(X) \lo C(Y)$ and $S: A(X,E) \lo C(Y,F)$ of linear operators, where $A(X)$ is a regular Banach function algebra on $X$, such that $f· g=0$ implies $Tf · Sg=0$, for all $f\in A(X)$ and $g\in A(X,E)$. If $T$ and $S$ are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between $X$ and $Y$ and, furthermore, $T^-1$ and $S^-1$ are also jointly separating maps.

Pourghobadi, Z., Tavani, M., Sady, F. 2020. Jointly separating maps between vector-valued function spaces. In Mathematica Slovaca, vol. 70, no.3, pp. 707-718. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0384

Pourghobadi, Z., Tavani, M., Sady, F. (2020). Jointly separating maps between vector-valued function spaces. Mathematica Slovaca, 70(3), 707-718. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0384

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava