On the Reflexivity of Non-selfadjoint Operator Algebras

Given an operator algebra $A$, we denote by $\operatorname{Lat} A$ its invariant subspace lattice. The algebra $A$ is called \emph{reflexive} if it coincides with the algebra of all operators leaving $\operatorname{Lat} A$ invariant. By von Neumann’s double commutant theorem, reflexive algebras may be viewed as a non-selfadjoint analogue of von Neumann algebras. Nest algebras, defined as those admitting a totally ordered invariant subspace lattice, were the first and remain the most studied example. Beyond totally ordered lattices, however, the structure of reflexive algebras becomes significantly subtler. 

In this talk, we focus on certain $w^{*}$-closed operator algebras on $L^{2}(\mathbb{R})$ generated by semigroups of translation, multiplication, and dilation operators. We discuss reflexivity results in this setting, consider structural features arising from the lack of projections or finite-rank generators, and, time permitting, comment on related questions for the associated norm-closed algebras.