Regularity for operator algebras on a Hilbert space

Abstract:

Four notions of regularity for operator algebras are introduced. An algebra $A$ is called $1$-regular if for any two linearly independent vectors $x,y \in H$ there is an $a \in A$ such that $ax = 0$ and $ay \ne 0$. We show that the only weakly closed transitive $1$-regular algebra is $B(H)$, thus providing a natural generalization of the Rickart-Yood density theorem. We construct an example of a $1$-regular algebra which contains no nonzero compact operators. This example is related to the "thin set" phenomena of classical harmonic analysis.