Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. B. Blackadar, K-Theory for Operator Algebras, Springer-Verlag, 1986. MR 88g:46082
  • 2. X. Dai and D.R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs A.M.S, to appear.
  • 3. J. Dixmier, Von Neumann algebras, North-Holland Pub. Comp., 1981. MR 83a:46004
  • 4. U. Haagerup and M. Rordam, Perturbations of the rotation C*-algebras and of the Heisenberg commutation relations, Duke Math. J. 77 (1995), 627-656. MR 96e:46073
  • 5. D. Han and V. Kamat, Operators and multiwaveles, preprint.
  • 6. R. V. Kadison, Representations of matricial operator algebras, Proc. Neptun Conf. on Op. Alg. and Group Rep. (1980), Pitman, 1984, Vol. 2, pp. 1-22. MR 85f:46104
  • 7. R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, Academic Press, 1986. MR 88d:46106
  • 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. Op. Theory 4 (1980), 201-210. MR 82d:46086
  • 10. M.A. Rieffel, C*-algebras associated with irrational rotations, Pac. J. Math 93 (1981), 415-429. MR 83b:46087

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46N99, 47N40, 47N99

Retrieve articles in all journals with MSC (1991): 46N99, 47N40, 47N99

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

Keywords: Irrational rotation unitary system, wandering vector and subspace
Received by editor(s): March 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society