## Taiwanese Journal of Mathematics

### RANK-ONE OPERATORS IN REFLEXIVE A-SUBMODULES OF OPERATOR ALGEBRAS

Dong Zhe

#### Abstract

In this paper, we first show that any reflexive $\mathcal{A}$-submodule $\mathcal{U}$ of a unital operator algebra $\mathcal{A}$ in $\mathcal{B(H)}$ is precisely of the following form: $$\mathcal{U} = \{ T \in \mathcal{B(H)}: TE \subseteq \phi(E) \quad \forall E \in \mathrm{Lat}\mathcal{A} \},$$ where $\phi$ is an order homomorphism of $\mathrm{Lat}\mathcal{A}$ into itself. Furthermore we investigate the density of the rank-one submodule of a reflexive $\mathcal{A}$-submodule in the $w^{*}$-topology and in certain pointwise approximation, and obtain several equivalent conditions by means of the order homomorphism $\phi$.

#### Article information

Source
Taiwanese J. Math., Volume 9, Number 3 (2005), 373-384.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407846

Digital Object Identifier
doi:10.11650/twjm/1500407846

Mathematical Reviews number (MathSciNet)
MR2162883

Zentralblatt MATH identifier
1103.47056

Subjects
Primary: 47L75: Other nonselfadjoint operator algebras

#### Citation

Zhe, Dong. RANK-ONE OPERATORS IN REFLEXIVE A-SUBMODULES OF OPERATOR ALGEBRAS. Taiwanese J. Math. 9 (2005), no. 3, 373--384. doi:10.11650/twjm/1500407846. https://projecteuclid.org/euclid.twjm/1500407846

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