Bimodules of Banach Space Nest Algebras

Abstract

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.

Lie Modules of Banach Space Nest Algebras

“…In this section, we resume the analysis of these order homomorphisms associated with bimodules. However, in contrast to [1,5,6], we obtain an explicit description of the order homomorphism solely determined by the rank one operators in the bimodule. This approach is akin to that taken in [8] in the Hilbert space setting.…”

“…It is well known that, for Banach and Hilbert spaces alike, weakly closed bimodules of nest algebras can be determined by order homomorphisms on the relevant nests (see, e.g., [1,5,6]). In this section, we resume the analysis of these order homomorphisms associated with bimodules.…”

“…Let A be a nest algebra having the π-property, and let L be a weakly closed Lie A-module. Then, there exists a weakly closed A-bimodule K, and there exists a weakly closed subalgebra D K of A such that L ⊆ K + D K , where K and D K are as in (6) and (11), respectively. Moreover, [K, A] ⊆ L.…”

“…Proof. Let K and D K be defined as in (6) and (11), respectively. By Lemma 6 (iii), we have [K, A] ⊆ L.…”

“…Associative, Jordan, and Lie ideals of nest algebras on Hilbert space have been substantially investigated in the literature (e.g., [1][2][3][4]), whilst in what concerns their module structure, or the corresponding theory in the Banach space setting, the literature is not so abundant. Notwithstanding, some headway in that direction has been made, as in the case of [5][6][7][8][9], for example.…”