Aaron R. Todd

Automatic continuity of linear maps on spaces of continuous functions

Let C(S)and C(T) denote the sup-normed Banach spaces of real- or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map A∶C(T)→C(S) is calledseparating if when two functions x and y from C(T) have disjoint cozero sets then so do Ax and Ay. In the spirit of [3] and [4], we show that separating maps are automatically continuous in some important cases (Theorems 2.4 and 2.5). If a separating map is continuous, then it must be a continuous multiple of a composition map (Theorem 2.2). If A is injective, separating and detaching (Def. 2.4) then S and T are homeomorphic (Theorem 2.1).

Barrelled spaces with(out) separable quotients

While the separable quotient problem is famously open for Banach spaces, in the broader context of barrelled spaces we give negative solutions. Obversely, the study of pseudocompact \(X\) and Warner bounded \(X\) allows us to expand Rosenthals positive solution for Banach spaces of the form \(C_{c}(X)\) to barrelled spaces of the same form, and see that strong duals of arbitrary \(C_{c}(X)\) spaces admit separable quotients. DOI: 10.1017/S0004972714000422

Oxtoby's pseudocompleteness revisited

Abstract It is well known that the class of Baire spaces is not productive, and the Baire Unification problem calls for finding subclasses that are productive and have “nice” inheritance properties. This paper extends a modified version of the pseudocompleteness property of J.C. Oxtoby using the bitopological spaces of J.C. Kelly, and considers its relation to the Baire property and (non) permanence properties including productivity and inheritance by G δ subsets of both.