Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Wandering vectors for
irrational rotation unitary systems


Author: Deguang Han
Journal: Trans. Amer. Math. Soc. 350 (1998), 309-320
MSC (1991): Primary 46N99, 47N40, 47N99
MathSciNet review: 1451604
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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.


References [Enhancements On Off] (What's this?)

  • 1. Bruce Blackadar, 𝐾-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867
  • 2. X. Dai and D.R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs A.M.S, to appear.
  • 3. Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
  • 4. Uffe Haagerup and Mikael Rørdam, Perturbations of the rotation 𝐶*-algebras and of the Heisenberg commutation relation, Duke Math. J. 77 (1995), no. 3, 627–656. MR 1324637, 10.1215/S0012-7094-95-07720-5
  • 5. D. Han and V. Kamat, Operators and multiwaveles, preprint.
  • 6. R. V. Kadison, Representations of matricial operator algebras, Operator algebras and group representations, Vol. II (Neptun, 1980), Monogr. Stud. Math., vol. 18, Pitman, Boston, MA, 1984, pp. 1–22. MR 733299
  • 7. Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186
  • 8. W. S. Li, J. E. McCarthy and D. Timotin, A note on wavelets for unitary systems, preprint.
  • 9. M. Pimsner and D. Voiculescu, Imbedding the irrational rotation 𝐶*-algebra into an AF-algebra, J. Operator Theory 4 (1980), no. 2, 201–210. MR 595412
  • 10. Marc A. Rieffel, 𝐶*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572

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Additional Information

Deguang Han
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
Email: D.Han@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02065-0
Keywords: Irrational rotation unitary system, wandering vector and subspace
Received by editor(s): March 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society