Operator algebras, wandering subspaces and wavelet theory /
We show that given a unitary system U that is closed under
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| Format: | Thesis Book |
| Language: | English |
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[Place of publication not identified] :
[publisher not identified] ;
1997.
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| Subjects: | |
| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=736579991&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | We show that given a unitary system U that is closed under right multiplication by elements of a group Uo contained in it, there exists a *-anti-homomorphism K [ ], (that depends on the wandering vector V)) that maps the von Neumann algebra w* (Uo) onto IS' (U) ( which is the intersection of the local commutant of U at 'O and the commutant of the projection Po on Ep = [UoO] ) such that for each unitary operator U in w* (Uo), K[ ] (U) is a unitary operator in C,, (U) with the property that Uo = K [ ], (U),O. This generalizes a result of Dai and Larson who proved it for the case where Uo is abelian. We also give a positive solution to an open problem posed by Dai and Larson. We show that even when Uo is non- abelian, and 71 is a wandering vector in E[ ] for some fixed wandering vector O, there exists a unitary operator W E w*(Uo) such that q = W[ ]. Then it is shown that the theory of unitary systems and wandering vectors can be generalized to the more general situation of wandering subspaces. We show that there is a *-anti-homomorphism from w*(Uo) into the local commutant of U at a wandering subspace p. In addition we show that if a unitary group has a wandering subspace of a certain dimension then it can have wandering subspaces of only that one dimension. We also show that the standard wavelet unitary system has wandering subspaces of any dimension including infinity. |
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| Item Description: | Vita. "Major Subject: Mathematics". |
| Physical Description: | vi, 56 leaves ; 28 cm. Issued also on microfiche from University Microfilms Inc. |
| Bibliography: | Includes bibliographical references: pages 54-55. |