Proceedings of the American Mathematical Society

Author(s): Michael T. Jury; David W. Kribs
Journal: Proc. Amer. Math. Soc. 133 (2005), 213-222.
MSC (2000): Primary 47A20, 47A45
Posted: June 23, 2004
Retrieve article in: PDF DVI PostScript

Abstract: Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

1.: W. Arveson, An invitation to $\mathrm{C}^*$ -algebras, Graduate Texts in Mathematics, No.39, Springer-Verlag, New York-Heidelberg, 1976. MR 58:23621
2.: J.A. Ball, V. Vinnikov, Functional models for representations of the Cuntz algebra, preprint, 2002.
3.: O. Bratteli and P.E.T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999) no.663. MR 99k:46094a
4.: J. Bunce, Models for n-tuples of non-commuting operators, J. Func. Anal. 57 (1984), 21-30. MR 85k:47019
5.: K.R. Davidson, E. Katsoulis, D.R. Pitts, The structure of free semigroup algebras, J. reine angew. Math. 533 (2001), 99-125. MR 2002a:47107
6.: K.R. Davidson, D.W. Kribs, M.E. Shpigel, Isometric dilations of non-commuting finite rank n-tuples, Can. J. Math. 53 (2001), 506-545. MR 2002f:47010
7.: 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
8.: M. Ephrem, Characterizing liminal and type I graph $\mathrm{C}^*$ -algebras, arXiv:math.OA/0211241, preprint, 2003.
9.: C. Foias, B. Sz. Nagy, Harmonic analysis of operators on Hilbert space, North Holland Pub. Co., London, 1970. MR 43:947
10.: A. Frazho, Models for non-commuting operators, J. Func. Anal. 48 (1982), 1-11. MR 84h:47010
11.: A. Frahzo, Complements to models for non-commuting operators, J. Func. Anal. 59 (1984), 445-461. MR 86h:47010
12.: J. Glimm, Type I C*-algebras, Math. Ann. 73 (1961), 572-612. MR 23:A2066
13.: F. Jaeck, S.C. Power, The semigroupoid algebras of finite graphs are hyper-reflexive, preprint, 2003.
14.: P.E.T. Jorgensen, Minimality of the data in wavelet filters, Adv. in Math. 159 (2001), 143-228. MR 2002h:46092
15.: M.T. Jury, D.W. Kribs, Ideal structure in free semigroupoid algebras from directed graphs, J. Operator Theory, to appear.
16.: E. Katsoulis, D.W. Kribs, Isomorphisms of algebras associated with directed graphs, preprint, 2003.
17.: D.W. Kribs, S.C. Power, Partly free algebras, Proc. International Workshop on Operator Theory and its Applications 2002, to appear.
18.: D.W. Kribs, S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc., to appear.
19.: D.W. Kribs, The curvature invariant of a non-commuting -tuple, Integral Eqtns. & Operator Thy., 41 (2001), 426-454. MR 2003b:47013
20.: A. Kumjian, D. Pask, I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math 184 (1998), 161-174. MR 99i:46049
21.: A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, Groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505-541. MR 98g:46083
22.: P.S. Muhly, A finite dimensional introduction to operator algebra, A. Katavolos (ed.), Operator Algebras and Applications, 313-354, Kluwer Academic Publishers, 1997. MR 98h:46062
23.: P.S. Muhly, B. Solel, Tensor algebras, induced representations, and the Wold decomposition, Can. J. Math. 51 (1999), 850-880. MR 2000i:46052
24.: P.S. Muhly, B. Solel, Tensor algebras over $\mathrm{C}^*$ -correspondences: representations, dilations, and $\mathrm{C}^*$ -envelopes, J. Func. Anal. 158 (1998), 389-457. MR 99j:46066
25.: G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), 523-536. MR 90c:47006
26.: G. Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001), 264-309. MR 2002b:46097
27.: J.J. Schaffer, On unitary dilations of contractions, Proc. Amer. Math. Soc. 6 (1955), 322. MR 16,934c

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A20, 47A45

Michael T. Jury
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: jury@math.purdue.edu

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS