Transitive spaces of operators

TL;DR

This paper explores the properties of transitive operator spaces, determining minimal dimensions for k-transitivity in finite dimensions, analyzing tensor products, and discussing infinite-dimensional analogues and related conjectures.

Contribution

It provides new results on minimal dimensions of k-transitive spaces, relations between transitivity of tensor products, and counterexamples to existing conjectures.

Findings

Minimal dimensions of k-transitive spaces in finite dimensions are determined.

Relations between transitivity of tensor products and their factors are established.

Counterexamples to some natural conjectures are presented.

Abstract

We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson's result on the weak-operator density of transitive spaces that are masa bimodules.