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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On frames for countably generated Hilbert $ C^*$-modules


Author: Ljiljana Arambasic
Journal: Proc. Amer. Math. Soc. 135 (2007), 469-478
MSC (2000): Primary 46L99; Secondary 46L05, 46H25
Published electronically: August 10, 2006
MathSciNet review: 2255293
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Abstract: Let $ V$ be a countably generated Hilbert $ C^*$-module over a $ C^*$-algebra $ A.$ We prove that a sequence $ \{f_i:i\in I\}\subseteq V$ is a standard frame for $ V$ if and only if the sum $ \sum_{i\in I}\langle x,f_i\rangle\langle f_i,x\rangle$ converges in norm for every $ x\in V$ and if there are constants $ C,D>0$ such that $ C\Vert x\Vert^2\le \Vert \sum_{i\in I}\langle x,f_i\rangle\langle f_i,x\rangle \Vert \le D\Vert x\Vert^2$ for every $ x\in V.$ We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert $ C^*$-modules over the $ C^*$-algebra of all compact operators on some Hilbert space is discussed.


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Additional Information

Ljiljana Arambasic
Affiliation: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
Email: ljsekul@math.hr

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08498-X
PII: S 0002-9939(06)08498-X
Keywords: $C^*$-algebra, Hilbert $C^*$-module, frame, frame transform, frame operator, compact operator
Received by editor(s): July 30, 2005
Received by editor(s) in revised form: September 19, 2005
Published electronically: August 10, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.