Jordan Modules and Jordan Ideals of Reflexive Algebras

Li, Pengtong; Li, Fengjie

doi:10.1007/s00020-012-1982-8

Cited by 3 publications

(3 citation statements)

References 19 publications

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“…Deguang showed in [1] that, given any reflexive subalgebra σweakly generated by its rank one operators, the σ-weakly closed bimodules over the algebra could analogously be characterized in terms of order homomorphisms on the lattice of invariant subspaces of the algebra. Li and Li [6,Proposition 2.6] have extended the mentioned results to the realm of Banach spaces. It is worth noticing that the bimodules considered in the Erdos-Power theorems are implicitly reflexive subspaces in the sense of Loginov-Shulman (cf.…”

Section: Introductionmentioning

confidence: 99%

A characterization of reflexive spaces of operators

Bračič

Oliveira

2017

Czech.Math.J.

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We show that for a linear space of operators M ⊆ B(H 1 , H 2 ) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ 1 , ψ 2 ) on a bilattice Bil(M) of subspaces determined by M, with P ≤ ψ 1 (P, Q) and Q ≤ ψ 2 (P, Q), for any pair (P, Q) ∈ Bil(M), and such that an operator T ∈ B(H 1 , H 2 ) lies in M if and only if ψ 2 (P, Q)T ψ 1 (P, Q) = 0 for all (P, Q) ∈ Bil(M). This extends to reflexive spaces the Erdos-Power type characterization of weakly closed bimodules over a nest algebra. 2010 Mathematics Subject Classification. Primary 47A15.

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Section: Introductionmentioning

confidence: 99%

A characterization of reflexive spaces of operators

Bračič

Oliveira

2017

Czech.Math.J.

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“…Although several results have already been generalized from the Hilbert space setting to that of Banach spaces (see, e.g., [10], [17], [18], [26]), that has not been the case in what concerns essential supports and their associated bimodules.…”

Section: Introductionmentioning

confidence: 99%

“…Finite rank operators play a decisive role in the theory of Hilbert space nest algebras and have been thoroughly investigated with respect to their density and decomposability (e.g., [12,19,20,22,23]). In fact, they have shown to be crucial in the characterisation of associative, Jordan and Lie modules of nest algebras on Hilbert space (e.g., [17,18,23,24]). Still, more recently, they have been given a prominent place in the related context of triangularizability (e.g., [6]).…”

Section: Introductionmentioning

confidence: 99%

Bimodules of Banach space nest algebras

Duarte¹,

Oliveira²

2020

Preprint

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We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra A, we charaterise the maximal and the minimal A-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal A-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions.

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Bimodules of Banach Space Nest Algebras

Duarte

Oliveira

2021

The Quarterly Journal of Mathematics

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We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long-established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal{A}$, we characterize the maximal and the minimal $\mathcal{A}$-bimodules having a given essential support function or support function pair. These characterizations are complete except for the minimal $\mathcal{A}$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions.

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Jordan Modules and Jordan Ideals of Reflexive Algebras

Cited by 3 publications

References 19 publications

A characterization of reflexive spaces of operators

A characterization of reflexive spaces of operators

Bimodules of Banach space nest algebras

Bimodules of Banach Space Nest Algebras

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