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Generalized Keller spaces. (English) Zbl 1169.46037

H.A.Keller in [Math.Z.172, No.1,41–49 (1980; Zbl 0414.46018)] constructed the first example of an infinite-dimensional orthomodular space. The author of the present paper constructs this Keller space, as well as some of the generalizations of it, as linear subspaces of formal power series fields.
The main result of the paper under review states the following. Let \(\rho\) be finite, \(\rho = n+1\), or a limit ordinal. Let \((\Lambda_{\xi})_{\xi < \rho}\) be a strictly increasing family of totally ordered abelian groups with the properties (g1)\(-\)(g3) considered in p. 981 of the paper. Let \(\Delta = \bigcup_{\xi < \rho} \Lambda_{\xi} \cup \{0 \}\) and let \(\Gamma \setminus \{0 \}= \sqrt{\Delta \setminus \{ 0 \}} \). Let \(F\) be any field with char \(F \neq 2\) and let \(H\) be the set all of \(f \in F^{\Gamma}\) for which its support, supp\((f)\), is dually well-ordered in the order of \(\Gamma \setminus \{0\}\). Let \(w: H \rightarrow \Gamma\) be the valuation \( f \rightarrow \max \text{supp}(f)\). Let \(K \subset H\) be the completion of the subfield \(F(\Delta \setminus \{ 0\})\) of \(H\). The field \(H\), with the norm \(\| f \| = w(f)\), is a normed space over \((K, w, \Delta)\). For \(\xi \in \rho^{*} = \{ \xi: \xi < \rho\} \cup \{ 0 \}\), put \(e_0 = 1\) and for \(\xi < \rho\) define \(e_{\xi} \in H\) by \(e_{\xi}(\gamma_{\xi}) =1\), \(e_{\xi}(\gamma) =0\) for \(\gamma \neq \gamma_{\xi}\). Let \(E \subset H\) be the completion of the linear span of \( \{e_{\xi} : \xi \in \rho^{*} \}\) over \(K\). Then \(E\) is, with the inner product defined as a bilinear and continuous extension of \(\langle e_{\xi}, e_{\xi}\rangle= e_{\xi}^{2}\) and \(\langle e_{\xi}, e_{\eta}\rangle=0\) for \(\xi \neq \eta\), a definite Banach space. \(\{ e_{\xi}: \xi \in \rho^{*}\}\) is an orthogonal basis for \(E\). If \(\rho = n +1\), the cofinal type of \(\Delta\) is equal to the cofinal type of \(\Lambda_n\), hence countable or not countable; if \(\rho\) is infinite, then \(\Delta\) has the cofinal type of \(\rho\).
The space \(E\) of the above result is called a generalized Keller space. Several properties of \(E\) are studied in the remainder of the paper, among them, when it is a form-Hilbert space.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
12J20 General valuation theory for fields
46C99 Inner product spaces and their generalizations, Hilbert spaces

Citations:

Zbl 0414.46018
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Full Text: Euclid