Transactions of the American Mathematical Society

Author(s): B. V. Rajarama Bhat; Tirthankar Bhattacharyya; Santanu Dey
Journal: Trans. Amer. Math. Soc. 356 (2004), 1551-1568.
MSC (2000): Primary 47A20, 47A13, 46L05, 47D25
Posted: October 6, 2003
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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.

1.: J. Agler, `The Arveson extension theorem and coanalytic models', Integral Equations and Operator Theory, 5 (1982) 608-631. MR 84g:47011
2.: A. Arias and G. Popescu, `Noncommutative interpolation and Poisson transforms,' Israel J. Math., 115 (2000) 205-234. MR 2001i:47021
3.: A. Arias and G. Popescu, `Noncommutative interpolation and Poisson transforms II', Houston J. Math., 25 (1999) 79-98. MR 2000i:47148
4.: W. B. Arveson, An Invitation to -algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 58:23621
5.: W. B. Arveson, `Subalgebras of -algebras III, Multivariable operator theory,' Acta Math., 181 (1998) 159-228. MR 2000e:47013
6.: A. Athavale, `On the intertwining of joint isometries,' J. Operator Theory, 23 (1990) 339-350. MR 91i:47029
7.: A. Athavale, `Model theory on the unit ball in $\mathbb{C}^m$ ', J. Operator Theory, 27 (1992) 347-358. MR 94i:47011
8.: B. V. Rajarama Bhat and T. Bhattacharyya, `A model theory for -commuting contractive tuples', J. Operator Theory 47 (2002) 97-116. MR 2003c:47018
9.: O. Bratteli and Palle E. T. Jorgensen Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc., 139 (1999), no. 663. MR 99k:46094a
10.: O. Bratteli and Palle E. T. Jorgensen, `Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale ', Integral Equations Operator Theory, 28 (1997) 382-443. MR 99k:46094b
11.: J. W. Bunce, `Models for -tuples of noncommuting operators,' J. Funct. Anal., 57 (1984) 21-30. MR 85k:47019
12.: J. Cuntz, `Simple -algebras generated by isometries,' Commun. Math. Phys., 57 (1977) 173-185. MR 57:7189
13.: C. 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 Funct. Anal. Appl., Kumasi, Ghana, 1979, pp. 159-182. MR 84e:47012
14.: K. R. Davidson, D. W. Kribs, and M. E. Shpigel, `Isometric dilations of non-commuting finite rank -tuples', Canad. J. Math., 53 (2001) 506-545. MR 2002f:47010
15.: Dey, S. `Standard dilations of -commuting tuples', Indian Statistical Institute, Bangalore preprint (2003).
16.: S. W. Drury, `A generalization of von Neumann's inequality to the complex ball', Proc. Amer. Math. Soc., 68 (1978) 300-304. MR 80c:47010
17.: A. E. Frazho, `Models for noncommuting operators', J. Funct. Anal., 48 (1982) 1-11. MR 84h:47010
18.: A. E. Frazho, `Complements to models for noncommuting operators', J. Funct. Anal., 59 (1984) 445-461. MR 86h:47010
19.: P. R. Halmos, A Hilbert Space Problem Book, Second Edition, Graduate Texts in Mathematics, No. 19, Springer-Verlag, New York-Berlin, 1982. MR 84e:47001
20.: S. Parrott, `Unitary dilations for commuting contractions', Pacific J. Math.. 34 (1970) 481-490. MR 42:3607
21.: G. Popescu, `Isometric dilations for infinite sequences of noncommuting operators', Trans. Amer. Math. Soc., 316 (1989) 523-536. MR 90c:47006
22.: G. Popescu, `Models for infinite sequences of noncommuting operators', Acta Sci. Math. (Szeged), 53 (1989) 355-368. MR 91b:47025
23.: G. Popescu, `Characteristic functions for infinite sequences of noncommuting operators', J. Operator Theory, 22 (1989) 51-71. MR 91m:47012
24.: G. Popescu, `Poisson transforms on some $C\sp *$ -algebras generated by isometries', J. Funct. Anal., 161 (1999) 27-61. MR 2000m:46117
25.: G. Popescu, `Curvature invariant for Hilbert modules over free semigroup algebras', Advances in Mathematics, 158 (2001) 264-309. MR 2002b:46097
26.: C. R. Putnam, Commutation properties of Hilbert space Operators and Related Topics, Springer-Verlag, New York, 1967. MR 36:707
27.: J. J. Schäffer, `On unitary dilations of contractions', Proc. Amer. Math. Soc., 6 (1955) 322. MR 16:934c
28.: B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam (1970). MR 43:947

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47A20, 47A13, 46L05, 47D25

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

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