---
tag: 672HNBV803
kind: concept
name: Reflexive operator algebra
msc: [47L05, 47A15]
sources:
  - url: https://en.wikipedia.org/wiki/Reflexive_operator_algebra
same_as:
  wikidata: Q7307359
  wikipedia: Reflexive operator algebra
---

## Summary

A reflexive operator algebra is an algebra of operators on a Hilbert space that is determined by its [[invariant subspaces|0ACX5QT8TE]].
It contains every operator that leaves all of them invariant.
Reflexivity links the algebraic structure of operator algebras with the lattice of their invariant subspaces.
Von Neumann algebras and nest algebras are reflexive, and the question of which algebras are reflexive is close to the [[invariant subspace problem|5RF055GK4N]].

## Explanations

### L1 human

Each set of operators has a family of subspaces that it maps into themselves.
Starting from these subspaces, one can collect all operators that respect them, and this collection contains the original algebra.
The algebra is reflexive when nothing new appears.
For example, the upper triangular $2 \times 2$ matrices are reflexive, while the matrices $\begin{pmatrix} a & b \\ 0 & a \end{pmatrix}$ are not, since they have the same invariant subspaces as all upper triangular matrices.