Disjointness preserving and local operators on algebras of differentiable functions (Q2731122)

The present paper is concerned with automatic continuity and representation theory of disjointness preserving operators on function spaces. Local operators are special cases. Given two linear spaces \(A (\Omega)\) and \(B (\Gamma)\) of complex-valued functions on the nonempty sets \(\Omega\) and \(\Gamma\), respectively, a linear mapping \(T : A (\Omega) \rightarrow B (\Gamma)\) is disjointness preserving provided \((T f) ( T g) = 0\) whenever \(f g = 0\). Of all the very interesting results let us mention two: NEWLINENEWLINENEWLINE1) Let \(\Omega\) be a normal \(k\)-space and let \(A (\Omega) \subset B (\Omega)\)be to normal subalgebras of \(C (\Omega) = \{ f \in \mathbb{C}^{\Omega} : f \) continuous \(\}\). Suppose that \(A (\Omega)\) and \(B (\Omega)\) are equipped with topologies such they are complete, metrizable topological algebras. Then \(A (\Omega)\) is continuously embeddable in \(C (\Omega)\). Moreover every local operator \(T\) from \(A (\Omega)\) to \(B (\Omega)\) is automatically continuous. NEWLINENEWLINENEWLINE2) Let \(\Omega \subset \mathbb{R}^n\) be an open set and \(\Gamma\) a locally compact Hausdorff space. Moreover let \(C^n (\Omega)\) be the space of functions for which all derivatives of order \(m\) exists and are continuous. Let \(T : C^n (\Omega) \rightarrow C_0 (\Gamma)\) be a disjointness preserving operator and set \(\Gamma_T = \{ \gamma \in \Gamma : \exists f \in C^m ( \Omega) ( T f (\gamma) \neq 0) \} \). Then there exists functions \(h_{\alpha} \in C_0 (\Gamma)\) for all multiindices \(\alpha\) with \(|\alpha |\leq m\) and a continuous mapping \(\rho : \Gamma_T \rightarrow \Omega\) such that NEWLINE\[NEWLINE(T f) ( \gamma) = \sum_{|\alpha |\leq m} h_{\alpha} (\gamma) ( D^{\alpha} f) ( \rho (\gamma))NEWLINE\]NEWLINE for all \(f \in C^m (\Omega)\), \(\gamma \in \Gamma_T\) This representation ist unique. NEWLINENEWLINENEWLINEThe well-known result due to \textit{J. Peetre} [Math. Scand. 8, 116-120 (1960; Zbl 0097.10402)] on the characterization of differential operators on \(C^m (\Omega)\) is a simple consequence of this theorem.