Wandering vectors for irrational rotation unitary systems

Abstract:

An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for wandering vector multipliers, and strengthen this by showing that for an irrational rotation unitary system $\mathcal {U}$, every unitary operator in $w^{*}(\mathcal {U})$ is a wandering vector multiplier. Moreover, we show that there is a class of wandering vector multipliers, induced in a natural way by pairs of characters of the integer group $\mathbb {Z}$, which fail to factor even as the product of a unitary in $\mathcal {U}’$ and a unitary in $w^{*}(\mathcal {U})$. Incomplete maximal wandering subspaces are also considered, and some questions are raised.