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Session 4. Banach Spaces and Operator Theory with Applications |
On the space of maximal ideals of vector lattices |
| Martin R. Weber, Technical University Dresden, Department of Mathematics, Germany |
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If the set \({\mathfrak M}(E)\) of all maximal ideals of an
Archimedean vector lattice \(E\) is equipped with the hull-kernel
topology \(\tau_{hk}\) then the topological space \(\big({\mathfrak
M}(E),\tau_{hk}\big)=:{\mathfrak M}(E)\) turns out to be a
Hausdorff space but, in general, does not satisfy any stronger
separation axiom. The space \({\mathfrak M}(E)\) carries many
information on the vector lattice \(E\). An element \(\varphi\in E\) is called finite if there exists an element \(z\in E\) (a majorant of \(\varphi\)) satisfying the property: for each \(x\in E\) there is a number \(c_x>0\) such that the inequality \[ |x|\wedge n |\varphi| \leq c_x z \] holds for any \(n\in \mathbb{N}\). The element \(\varphi\) is called totally finite, if it possesses a majorant which itself is a finite element. A finite element \(\varphi\) is called selfmajorizing, if \(|\varphi|\) is a majorant of \(\varphi\). Finite, totally finite and selfmajorizing elements of a vector lattice \(E\) can be characterized by means of the subsets \(G_x:=\{M\in {\mathfrak M}(E): x\notin M\}\) which are defined in \({\mathfrak M}(E)\) for any \(x\in E\). The \(\tau_{hk}\)-closure \(\operatorname{supp}_\mathfrak{M}(x)\) of \(G_x\) is the so-called abstract support of the element \(x\). In radical-free vector lattices a finite element \(\varphi\) is characterized by the compactness of its abstract support and the majorants \(z\) of \(\varphi\) by the inclusion \({\rm supp}_\mathfrak{M}(\varphi)\subset G_z\). Totally finite and selfmajorizing elements can be similarly characterized.
The space \({\mathfrak M}(E)\) plays an important role for the
representation of vector lattices by means of continuous functions,
namely in the situation if one asks for representations, where the
isomorphic images of finite elements are functions with compact support. |
References
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