Standard noncommuting and commuting dilations of commuting tuples
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- by B. V. Rajarama Bhat, Tirthankar Bhattacharyya and Santanu Dey PDF
- Trans. Amer. Math. Soc. 356 (2004), 1551-1568 Request permission
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.References
- Jim Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 5, 608–631. MR 697007, DOI 10.1007/BF01694057
- Alvaro Arias and Gelu Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234. MR 1749679, DOI 10.1007/BF02810587
- Alvaro Arias and G. Popescu, Noncommutative interpolation and Poisson transforms. II, Houston J. Math. 25 (1999), no. 1, 79–98. MR 1675377
- William Arveson, An invitation to $C^*$-algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 0512360
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- Ameer Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (1990), no. 2, 339–350. MR 1066811
- Ameer Athavale, Model theory on the unit ball in $\textbf {C}^m$, J. Operator Theory 27 (1992), no. 2, 347–358. MR 1249651
- B. V. Rajarama Bhat and Tirthankar Bhattacharyya, A model theory for $q$-commuting contractive tuples, J. Operator Theory 47 (2002), no. 1, 97–116. MR 1905815
- Ola Bratteli and Palle E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, x+89. MR 1469149, DOI 10.1090/memo/0663
- Ola Bratteli and Palle E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, x+89. MR 1469149, DOI 10.1090/memo/0663
- John W. Bunce, Models for $n$-tuples of noncommuting operators, J. Funct. Anal. 57 (1984), no. 1, 21–30. MR 744917, DOI 10.1016/0022-1236(84)90098-3
- Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330
- Chandler Davis, Some dilation and representation theorems, Proceedings of the Second International Symposium in West Africa on Functional Analysis and its Applications (Kumasi, 1979) Forum for Functional Anal. Appl., Kumasi, Ghana, 1979, pp. 159–182. MR 698808
- Kenneth R. Davidson, David W. Kribs, and Miron E. Shpigel, Isometric dilations of non-commuting finite rank $n$-tuples, Canad. J. Math. 53 (2001), no. 3, 506–545. MR 1827819, DOI 10.4153/CJM-2001-022-0
- Dey, S. ‘Standard dilations of $q$-commuting tuples’, Indian Statistical Institute, Bangalore preprint (2003).
- S. W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300–304. MR 480362, DOI 10.1090/S0002-9939-1978-0480362-8
- Arthur E. Frazho, Models for noncommuting operators, J. Functional Analysis 48 (1982), no. 1, 1–11. MR 671311, DOI 10.1016/0022-1236(82)90057-X
- Arthur E. Frazho, Complements to models for noncommuting operators, J. Funct. Anal. 59 (1984), no. 3, 445–461. MR 769375, DOI 10.1016/0022-1236(84)90059-4
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
- Stephen Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970), 481–490. MR 268710
- Gelu Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. MR 972704, DOI 10.1090/S0002-9947-1989-0972704-3
- Gelu Popescu, Models for infinite sequences of noncommuting operators, Acta Sci. Math. (Szeged) 53 (1989), no. 3-4, 355–368. MR 1033608
- Gelu Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), no. 1, 51–71. MR 1026074
- Gelu Popescu, Poisson transforms on some $C^*$-algebras generated by isometries, J. Funct. Anal. 161 (1999), no. 1, 27–61. MR 1670202, DOI 10.1006/jfan.1998.3346
- Gelu Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001), no. 2, 264–309. MR 1822685, DOI 10.1006/aima.2000.1972
- C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR 0217618
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Additional Information
- B. V. Rajarama Bhat
- Affiliation: Indian Statistical Institute, R. V. College Post, Bangalore 560059, India
- MR Author ID: 314081
- 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
- Received by editor(s): December 10, 2002
- Received by editor(s) in revised form: February 20, 2003
- Published electronically: October 6, 2003
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2034318