2-universally complete Riesz spaces and inverse-closed Riesz spaces (Q2503004)

Let \(X\) be a nonempty set. The first result of the authors is about uniformly closed Riesz subspaces of \(\mathbb{R}^X\) containing the constant functions. Theorem. Let \(E\) be a Riesz subspace of \(\mathbb{R}^X\) which is uniformly closed and contains the constants. Then the following are equivalent: (i) \(E\) is closed under inversion in \(X\). (ii) \(E\) is closed under composition with continuous functions defined on open intervals of \(\mathbb{R}\). (iii) \(E\) is closed under composition with a continuous function \(\varphi\) defined on an open interval \((a,b)\), \(a \in \mathbb{R}\), which is not bounded on any neighbourhood of \(a\). A further characterization of subspaces \(E\) of \(\mathbb{R}^X\) which are closed under inversion in \(X\) can be found in Theorem 2.9 of the paper. Let \(E\) be an Archimedean Riesz space with weak order unit \(e > 0\). \(\text{Spec}\,E\) denotes the Riesz homomorphisms in \(\mathbb{R}^{E}\) that carry \(e\) to~1, endowed with the product topology of \(\mathbb{R}^{E}\). This topology is determined by functions \(\widehat{f}:\text{Spec}\,E\rightarrow\mathbb{R}\), \(\widehat{f}(z) = z (f)\) \((f \in E)\). The mapping \(f \mapsto\widehat{f}\) of \(E\) into \(C(\text{Spec}\,E)\) is a lattice homomorphism that maps \(e\) to~1. \(E\) is called \(e\)-semisimple if the map \(f \mapsto\widehat{f}\) of \(E\) into \(C(\text{Spec}\,E\)) is injective. The image of \(E\) is a Riesz subspace of \(C(\text{Spec}\,E\)) when \(E\) is semisimple. The last result of the paper states that if \(E\) is an Archimedean Riesz space with a weak order unit \(e\) which is \(e\)-semisimple and \(e\)-uniformly complete, then \(E\) is 2-universally \(e\)-complete iff \(\widehat{E}\) is closed under inversion in \(\text{Spec}\,E\).