Norms of vector functionals

TL;DR

This paper investigates conditions under which the norm of vector functionals on operator algebras can be bounded using the algebra's invariant subspace lattice, introducing a new property satisfied by von Neumann and CSL algebras.

Contribution

It introduces a novel operator algebraic property linking vector functional norms to invariant subspace lattices and characterizes classes of algebras satisfying this property.

Findings

All von Neumann algebras satisfy the property.

All CSL algebras satisfy the property.

Some operator algebras do not satisfy the property or any scaled version.

Abstract

We examine the question of when, and how, the norm of a vector functional on an operator algebra can be controlled by the invariant subspace lattice of the algebra. We introduce a related operator algebraic property, and show that it is satisfied by all von Neumann algebras and by all CSL algebras. We exhibit examples of operator algebras that do not satisfy the property or any scaled version of it.