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 , every unitary operator in 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 , which fail to factor even as the product of a unitary in and a unitary in . Incomplete maximal wandering subspaces are also considered, and some questions are raised.
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Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Address at time of publication: Department of Mathematics, Qufu Normal University, Shandong, 273165 P.R. China
Keywords: Irrational rotation unitary system, wandering vector and subspace
Received by editor(s): March 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society