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Dugundji extenders and retracts on generalized ordered spaces. (English) Zbl 0919.54010

Let \(C(X)\) denote the ring of continuous real-valued functions on a Tikhonov space \(X\), and let \(A\) be a subspace of \(X\). A linear map \(\varphi :C(A)\to C(X)\) such that \(\varphi(f)| X=f\) is called an \(L_{\text{ch}}\)-extender (resp. \(L_{\text{cch}}\)-extender) if \(\varphi(f)[X]\) is contained in the convex hull (resp, closed convex hull) of \(f[A]\) for each \(f\in C(A)\).
This paper investigates a cluster of conditions related to existence of an \(L_{\text{ch}}\)-, or \(L_{\text{cch}}\)-extender from \(C(A)\) to \(C(X)\), where \(A\) is a closed subspace of the GO-space \(X\). Sample result:
The following are equivalent for a closed subset \(A\) of a GO-space \(X\): (a) \(A\) is a retract of the union of \(A\) and all clopen convex components of \(X\smallsetminus A\). (b) There is a continuous \(L_{\text{ch}}\)-extender \(\varphi: C(A\times Y)\to C(X \times Y)\), with respect to both the compact-open topology and the pointwise convergence topology, for each space \(Y\). (c) \(A\times Y\) is \(C^*\)-embedded in \(X\times Y\) for each space \(Y\). (d) There is a continuous linear extender \(\varphi: C^*_k(A)\to C_p(X)\).

MSC:

54C15 Retraction
54C20 Extension of maps
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54B10 Product spaces in general topology
54C45 \(C\)- and \(C^*\)-embedding
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