Abstract
We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of ℓ2. Examples are given whereA differs from ℱ (ℱ being the rank one subalgebra ofA), and where ℱ together with the identity I is not strongly dense inA. 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 ℱ. We show that this holds provided that I is approximable by compact operators inA at any two points of ℓ2. We show that the spaceA+ℒ* (where ℒ is the annihilator of ℱ) is ultraweakly dense in ℬ(ℓ2), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.
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Katavolos, A., Lambrou, M.S. & Papadakis, M. On some algebras diagonalized by M-bases of ℓ2 . Integr equ oper theory 17, 68–94 (1993). https://doi.org/10.1007/BF01322547
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DOI: https://doi.org/10.1007/BF01322547