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Two families of self-adjoint indecomposable operators in an orthomodular space. (English) Zbl 1163.47061

The non-existence of infinite-dimensional Hilbert spaces over non-archimedean valued fields is well-known. Surprisingly, if we allow the scalar field to have a more general so-called Krull valuation, Hilbert-like spaces appear. They are the so-called orthomodular spaces, that is, those vector spaces \(E\) over a Krull valued field \(K\) with a hermitian \(K\)-valued form \(\Phi\) on \(E \times E\) such that, for each subspace \(D\) of \(E\), one verifies \(D^{\perp\perp}=D\Rightarrow E=D\oplus D^{\perp}\). The first example of such an \(E\) was constructed by H.A.Keller in [Math.Z.172, No.1,41–49 (1980Zbl 0414.46018)]; we give a brief outline of its construction (see also Section 1 of the paper). Let \(\Gamma=\bigoplus_{j \in \mathbb{N}} \Gamma_j\), where each \(\Gamma_j\) is an isomorphic copy of the additive group of integers. \(\Gamma\) is ordered antilexicographically. Let \(K_0\) be the field of all rational functions in the variables \(X_1, X_2,\dots\) with real coefficients. Let us consider the Krull valuation \(K_0 \rightarrow \Gamma\cup\{\infty\}\) which is trivial on \(\mathbb{R}\) and sends each \(X_i\) to \((0,\dots,0,-1,0,\dots)\) (where the \(-1\) is in the \(i\)-th place). Let \(K\) be the completion of \(K_0\), valued with the corresponding (unique) extension to \(K\) of the valuation on \(K_0\). Then Keller’s orthomodular space is the \(K\)-vector space
\[ E=\bigg\{(\xi)_{i\in\mathbb{N}_0}\in K^{\mathbb{N}_0}:\sum_{i=0}^{\infty}\xi_{i}^{2}X_i \text{ converges in }K\bigg\} \]
with componentwise operations, endowed with the hermitian form \(\Phi:E\times E\rightarrow K\) defined by
\[ \Phi((\xi)_{i \in \mathbb{N}_0}, (\eta)_{i \in \mathbb{N}_0}) = \sum_{i=0}^{\infty}\xi_i\eta_iX_i \]
and with the natural non-archimedean norm \(E\rightarrow\Gamma \cup \{ \infty \}\) associated to \(\Phi\).
Now, let \({\mathcal B}(E)\) be the algebra of all linear operators \(B:E\rightarrow E\) for which the set \(\{\|B(x)\|-\|x\|:x\in E\setminus\{0\}\}\) is bounded from below in \(\Gamma\). By \(\{e_i:i\in\mathbb{N}_0\}\) we denote the canonical base of \(E\) (i.e., \(e_i=(0,0,\dots,0,1,0,\dots)\), where the \(1\) is in the \((i+1)\)-th place).
In Section 2, the author summarizes all the geometric properties of \(E\) and all the results related to \({\mathcal B}(E)\), proved by H. Keller and H. Ochsenius in previous papers, that will be needed in the next sections of the present work.
Section 3 contains the core of the work. Here, the author constructs two infinite families \({\mathcal F}_1\) and \({\mathcal F}_2\) in \({\mathcal B} (E)\) in such a way that each element \(B\) of these families is selfadjoint (i.e., \(\Phi(B(e_i), e_j) = \Phi(e_i,B(e_j))\) for all \(i,j\in\mathbb{N}_0\)), is indecomposable (i.e., \(B\) does not admit closed invariant subspaces of \(E\) apart from \(\{0\}\) and \(E\)) and its spectrum (i.e., \(\{\lambda\in K:B-\lambda I\text{ has no inverse in }{\mathcal B}(E)\}\)) is nonempty (as usual, \(I\) is the identity map on \(E\)).
Finally, in Section 4, given a \(B_1 \in {\mathcal F}_1\) (resp., a \(B_2 \in {\mathcal F}_2\)), the author considers the subalgebras of \({\mathcal B}(E)\) formed by the elements of \({\mathcal B}(E)\) that commute with \(B_1\) (resp., with \(B_2\)). She proves that these two subalgebras are mutually distinct and, on the other hand, that the intersection of \({\mathcal F}_1\) (resp., \({\mathcal F}_2\)) with a certain commutant subalgebra \({\mathcal A}\) of \({\mathcal B}(E)\) is \(\{\alpha I:\alpha\in K\}\).

MSC:

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47L10 Algebras of operators on Banach spaces and other topological linear spaces

Citations:

Zbl 0414.46018
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References:

[1] Barrios Rodríguez, Carla, Dos familias de operadores autoadjuntos e indescomponibles en un espacio ortomodular (2004)
[2] Gross, Herbert; Künzi, Urs-Martin, On a class of orthomodular quadratic spaces, Enseign. Math. (2), 31, 3-4, 187-212 (1985) · Zbl 0603.46030
[3] Keller, Hans A.; Ochsenius A., Hermina, Bounded operators on non-Archimedian orthomodular spaces, Math. Slovaca, 45, 4, 413-434 (1995) · Zbl 0855.46049
[4] Keller, Hans A.; Ochsenius A., Herminia, \(p\)-adic functional analysis (Nijmegen, 1996), 192, 253-264 (1997) · Zbl 0892.47074
[5] Keller, Hans Arwed, Ein nicht-klassischer Hilbertscher Raum, Math. Z., 172, 1, 41-49 (1980) · Zbl 0414.46018 · doi:10.1007/BF01182777
[6] Ribenboim, Paulo, Théorie des valuations, 1964 (1968) · Zbl 0139.26201
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