The Wickstead problem on Dedekind \(\sigma \)-complete vector lattices
The question of whether a band-preserving linear operator on an Archimedean vector lattice is automatically order-bounded was raised by \textit{A. W. Wickstead} [Compos. Math. 35, 225--238 (1977; Zbl 0381.47021)]. Later, Bernau, McPolin and Wickstead, and de Pagter, respectively, proved that if \(T\) is a band-preserving linear operator on an Archimedean vector lattice \(A\) and if for every positive sequence in \(A\) which converges to zero relatively uniformly, we have \(\inf_n\{|T(x_n)|\}= 0\), then \(T\) is order-bounded. Recall that if \(A\) is a universally complete vector lattice, a linear topology \(\tau\) on \(A\) is called an \(f\)-compatible topology if \(x_ny_n\rightarrow 0\) in \(\tau\) whenever \(x_n\) in \(\tau\) and \(y_n \rightarrow 0\) relatively uniformly. As the main result of the paper, the authors prove the following theorem. Theorem. Let \(A\) be a \(\sigma\)-complete Dedekind vector lattice, \(A^u\) be its universal completion, \(\tau\) be an \(f\)-compatible topology on \(A^u\) and \(T:A\rightarrow A\) be a band-preserving operator such that \(i\circ T:( A,(r.u\text{-topology}))\rightarrow (A^u,\tau)\) is continuous, then \(T\) is order-bounded. Similar results are obtained for Freudenthal vector lattices and bilinear maps. Finally, a new version of \textit{A. E. Gutman, A. G. Kusraev} and \textit{S. S. Kutateladze}'s result on the Wickstead problem [``The Wickstead problem'', Sib. Èlektron. Mat. Izv. 5, 293--333 (2008)] is given.
scientific article; zbMATH DE number 5692392
scientific article