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Relative annihilators and relative commutants in non‐selfadjoint operator algebras
Author(s) -
Marcoux L. W.,
Sourour A. R.
Publication year - 2012
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdr065
Subject(s) - mathematics , pure mathematics , commutative property , characterization (materials science) , distributive property , operator algebra , operator (biology) , annihilator , hilbert space , compact operator , centralizer and normalizer , discrete mathematics , algebra over a field , extension (predicate logic) , computer science , physics , biochemistry , chemistry , repressor , transcription factor , gene , programming language , optics
We extend von Neumann's Double Commutant Theorem to the setting of non‐selfadjoint operator algebras , while restricting the notion of commutants of a subset of to those operators in that commute with every operator in . If is a completely distributive commutative subspace lattice algebra acting on a Hilbert space ℋ, then we obtain an alternate characterization (to those of Erdos–Power and of Deguang) of the weak‐operator closed ideals of . In the case of nest algebras, we use this characterization to formulate an explicit characterization of the relative (double) commutants and relative (double) annihilators of these ideals. We also describe a property of subspaces of the algebra for which the relative commutants can be expressed as an extension of the relative annihilator by the scalar operators.