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On some algebras diagonalized by \(M\)-bases of \(\ell^ 2\). (English) Zbl 0796.47033

Summary: We study reflexive algebras \({\mathcal A}\) whose invariant lattices \(\text{Lat } {\mathcal A}\) are generated by \(M\)-bases of \(\ell^ 2\). Examples are given where \(\text{Lat } {\mathcal A}\) differs from \(\text{Lat } {\mathcal F}\) (\({\mathcal F}\) being the rank one subalgebra of \({\mathcal A}\)), and where \({\mathcal F}\) together with the identity \(I\) is not strongly dense in \({\mathcal A}\). For \(M\)-bases in a special class, we characterize the cases when they are strong, and also when the identity \(I\) is the ultraweak limit of a sequence of contractions in \({\mathcal F}\). We show that this holds provided that \(I\) is approximable by compact operators in \({\mathcal A}\) at any two points of \(\ell^ 2\). We show that the space \({\mathcal A}+ {\mathcal S}\) (where \({\mathcal S}\) is the annihilator of \({\mathcal F}\)) is ultraweakly dense in \({\mathcal B}(\ell^ 2)\), and characterize the \(M\)- bases in this class for which the sum is direct. We give a class of automorphisms of \({\mathcal A}\) which are strongly continuous but not spatial.

MSC:

47L30 Abstract operator algebras on Hilbert spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47L45 Dual algebras; weakly closed singly generated operator algebras
47L05 Linear spaces of operators
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