Disjointness preserving Fredholm operators in ultrametric spaces of continuous functions (Q2469096)

Let \(X,Y\) be \(\mathbb{N}\)-compact topological spaces, let \(K\) be a complete non-archimedean valued field, and let \(C(X)\), \(C(Y)\) be the spaces of all \(K\)-valued continuous functions on \(X\), \(Y\) respectively. In previous papers, the author gave some representations of the maps \(T : C(X) \rightarrow C(Y)\) that are disjointness preserving, i.e., \((T f) (T g) =0\) whenever \( fg =0\) (\(f,g \in C(X)\)). In the present paper, he describes those maps \(T\) that are disjointness preserving and Fredholm (a linear map between vector spaces over \(K\) is called Fredholm if its kernel is finite-dimensional and its image is finite-codimensional). This is the non-archimedean counterpart of the study carried out in the real and complex settings in [\textit{J.-H. Jeang} and \textit{N.-C. Wong}, J. Oper. Theory 49, No. 1, 61--75 (2003; Zbl 1030.46059)], although the techniques used in the paper under review are independent of those used in the classical case. The author proves that any disjointness preserving Fredholm map \(T: C(X) \rightarrow C(Y)\) satisfies six conditions, most of them related to properties of the set \(D := \bigcup \{ c(Tf) : f \in C(X) \}\) (where \(c(Tf):= \{ y \in Y : (Tf)(y) \neq 0 \}\), \(f \in C(X)\)) and of the support map of \(T\) (which is a function on \(D\) with values in the Banaschewski compactification of \(X\)). As in his previous works about disjointness preserving maps, now the author obtains also a version of the above for disjointness preserving Fredholm maps \(S: C^{*}(X) \rightarrow C^{*}(Y)\), where \(C^{*}(X)\), \(C^{*}(Y)\) are the spaces of all bounded \(K\)-valued continuous functions on \(X,Y\), respectively. In fact, such a map \(S\) again satisfies six conditions; the first five ones coincide with the ones required for the case of maps \(T: C(X) \rightarrow C(Y)\), but the sixth one differs slightly. It would be interesting, for future research work, to investigate whether the six conditions considered in this paper can be used to characterize the disjointness preserving Fredholm maps on the corresponding spaces of (bounded and not necessarily bounded) continuous functions.