Wandering Vectors for Irrational Rotation Unitary Systems

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 U , every unitary operator in w∗(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 Z, which fail to factor even as the product of a unitary in U ′ and a unitary in w∗(U). Incomplete maximal wandering subspaces are also considered, and some questions are raised. An important class of operator algebras is the class of irrational rotation C*algebras, which has been systematically studied over the past 15 years. These algebras have several equivalent definitions (see [10]). One is that they are exactly the C*-algebras Aθ generated by a pair of unitary elements u and v which satisfy the relation uv = λvu, where λ = exp(2πiθ) and θ ∈ (0, 1) is an irrational number. We will call the set U = {unvm : (n,m) ∈ Z × Z} an (abstract) irrational rotation unitary system, where Z is the set of all integers. It is a proper subset of the group generated by u and v. If B is a C*-algebra and a, b are two elements in B satisfying the relation ab = exp(2πiθ)ba, then it is known that there is a faithful *-isomorphism π from Aθ into B satisfying π(u) = a and π(v) = b (see [4] or [9]). Following Dai and Larson [2], a unitary system U is a subset of the unitary operators acting on a separable Hilbert space H which contains the identity operator. A norm one element ψ ∈ H is called a wandering vector for U if Uψ = {Uψ : U ∈ U} is an orthonormal set; that is, 〈Uψ, V ψ〉 = 0 if U, V ∈ U and U 6= V . If Uψ is an orthonormal basis for H , then ψ is called a complete wandering vector for U . The set of all complete wandering vectors for U is denoted by W(U). More generally, a closed subspace M of H is called a wandering subspace of U if UM and VM are orthogonal for any different U and V in U . A wandering subspace M is called complete if span{UM} = H . The set of all the complete wandering subspaces for U is denoted by WS(U). More generally, a unital unitary subset U of a C*-algebra A is called an abstract unitary system. In this case, one is interested in representations π of A for which π(U) has wandering subspaces. Received by the editors March 11, 1996. 1991 Mathematics Subject Classification. Primary 46N99, 47N40, 47N99.