# Free multivariate w*-semicrossed products: reflexivity and the bicommutant property

## TL;DR

This paper investigates the reflexivity and bicommutant property of w*-semicrossed products arising from actions of free semigroups and automorphisms on w*-algebras, establishing conditions for these properties.

## Contribution

It establishes reflexivity and bicommutant properties of w*-semicrossed products under various dynamics, extending previous results and providing new conditions for these properties.

## Key findings

- w*-semicrossed products are reflexive when dynamics are implemented by bounded invertible row operators
- w*-semicrossed products over factors are reflexive
- w*-semicrossed products have the bicommutant property if and only if the ambient algebra does

## Abstract

We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products over factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bicommutant property if and only if so does the ambient algebra of the dynamics.