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Standard noncommuting and commuting dilations of commuting tuples


Authors: B. V. Rajarama Bhat, Tirthankar Bhattacharyya and Santanu Dey
Journal: Trans. Amer. Math. Soc. 356 (2004), 1551-1568
MSC (2000): Primary 47A20, 47A13, 46L05, 47D25
DOI: https://doi.org/10.1090/S0002-9947-03-03310-5
Published electronically: October 6, 2003
MathSciNet review: 2034318
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Abstract: We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction, there are two commonly used dilations in multivariable operator theory. First there is the minimal isometric dilation consisting of isometries with orthogonal ranges, and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of the Cuntz algebra $\mathcal{O}_n$ coming from dilations of commuting tuples.


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Additional Information

B. V. Rajarama Bhat
Affiliation: Indian Statistical Institute, R. V. College Post, Bangalore 560059, India
Email: bhat@isibang.ac.in

Tirthankar Bhattacharyya
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
Email: tirtha@math.iisc.ernet.in

Santanu Dey
Affiliation: Indian Statistical Institute, R. V. College Post, Bangalore 560059, India
Email: santanu@isibang.ac.in

DOI: https://doi.org/10.1090/S0002-9947-03-03310-5
Keywords: Dilation, commuting tuples, complete positivity, Cuntz algebra
Received by editor(s): December 10, 2002
Received by editor(s) in revised form: February 20, 2003
Published electronically: October 6, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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