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Spectral synthesis and masa-bimodules. (English) Zbl 1038.47044

Let \({\mathcal H}_1\) and \({\mathcal H}_2\) be separable complex Hilbert spaces and \({\mathcal P}_i\) be the lattice of all (orthogonal) projections on \({\mathcal H}_i\), \(i= 1,2\). A map \(\phi:{\mathcal P}_1\to{\mathcal P}_2\) is called a subspace map of it in case it is \(0\)-preserving and \(\vee\)-continuous. Given a subspace \({\mathcal U}\subseteq{\mathcal B}({\mathcal H}_1,{\mathcal H}_2)\), we define its map \(\text{Map\,}{\mathcal U}:{\mathcal P}_1\to{\mathcal P}_2\) by \(\text{Map\;}{\mathcal U}(P)= \overline{[{\mathcal U}(P)]}\) \((P\in{\mathcal P}_1)\), where \(\overline{[{\mathcal U}(P)]}\) denotes the projection onto the closed subspace spanned by \(\{Sx: x\in P ({\mathcal H}_1),S\in{\mathcal U}\}\). A masa-bimodule \({\mathcal U}\subseteq{\mathcal B}({\mathcal H}_1,{\mathcal H}_2)\) is a subspace of operators, such that \({\mathcal D}_2{\mathcal U}{\mathcal D}_1\subseteq{\mathcal U}\) for some maximal abelian selfadjoint algebras (masas) \({\mathcal D}_1\subseteq{\mathcal B}({\mathcal H}_1)\) and \({\mathcal D}_2\subseteq{\mathcal B}({\mathcal H}_2)\). If \(\phi\) is a commutative subspace map, we consider the family \[ \Phi= \{{\mathcal U}\subseteq{\mathcal B}({\mathcal H}_1,{\mathcal H}_2):{\mathcal U}\text{ is a weak}^*\text{-closed }{\mathcal D}_2,\,{\mathcal D}_1\text{-bimodule,\,Map\,}{\mathcal U}= \phi\}. \] The map \(\phi\) is called synthetic if \(\Phi\) contains only one element. A reflexive masa-bimodule \({\mathcal M}\) is called synthetic if \(\text{Map\,}{\mathcal M}\) is synthetic. A commutative subspace map \(\phi\) is called a map of finite width if there exist nest maps \(\chi_1,\chi_2,\dots,\chi_n\) such that the (respective) semilattices of \(\chi_i\) commute with each other and \(\phi= \chi_1\wedge\chi_2\wedge\cdots\wedge \chi_n\).
The main result of the paper is the following: any map of finite width is synthetic. The author proves that there exists a non-synthetic thin masa-bimodule. The case of the multiplication masas on \(L^2(X,m)\) and \(L^2(Y,n)\) for measure spaces is also studied.

MSC:

47L05 Linear spaces of operators
47L35 Nest algebras, CSL algebras
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