On wandering vector multipliers for unitary groups

Authors:
Guoxing Ji and Kichi-Suke Saito

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3263-3269

MSC (2000):
Primary 46L10, 46L51

Published electronically:
May 2, 2005

MathSciNet review:
2161148

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Abstract: A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system acting on a separable Hilbert space into itself. It is proved that the wandering vector multipliers for a unitary group form a group, which gives a positive answer for a problem of Han and Larson. Furthermore, non-abelian unitary groups of order 6 are considered. We prove that the wandering vector multipliers of such a unitary group can not generate . This negatively answers another of their problems.

**1.**Xingde Dai and David R. Larson,*Wandering vectors for unitary systems and orthogonal wavelets*, Mem. Amer. Math. Soc.**134**(1998), no. 640, viii+68. MR**1432142**, 10.1090/memo/0640**2.**Deguang Han,*Wandering vectors for irrational rotation unitary systems*, Trans. Amer. Math. Soc.**350**(1998), no. 1, 309–320. MR**1451604**, 10.1090/S0002-9947-98-02065-0**3.**Deguang Han and D. Larson,*Wandering vector multipliers for unitary groups*, Trans. Amer. Math. Soc.**353**(2001), no. 8, 3347–3370. MR**1828609**, 10.1090/S0002-9947-01-02795-7**4.**Deguang Han and David R. Larson,*Frames, bases and group representations*, Mem. Amer. Math. Soc.**147**(2000), no. 697, x+94. MR**1686653**, 10.1090/memo/0697**5.**Thomas W. Hungerford,*Algebra*, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR**600654****6.**Richard V. Kadison,*Isometries of operator algebras*, Ann. Of Math. (2)**54**(1951), 325–338. MR**0043392****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.**I. E. Segal,*A non-commutative extension of abstract integration*, Ann. of Math. (2)**57**(1953), 401–457. MR**0054864**

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Additional Information

**Guoxing Ji**

Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xian, 710062, People’s Republic of China

Email:
gxji@snnu.edu.cn

**Kichi-Suke Saito**

Affiliation:
Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan

Email:
saito@math.sc.niigata-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-05-07860-3

Keywords:
Unitary group,
wandering vector,
wandering vector multiplier,
von Neumann algebra.

Received by editor(s):
April 30, 2003

Received by editor(s) in revised form:
June 10, 2004

Published electronically:
May 2, 2005

Additional Notes:
The first author was supported in part by the National Natural Science Foundation of China (No.10071047), the Excellent Young Teachers Program of MOE, P.R.C and China Scholarship Council. The second author was supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science.

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society