Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Katsoulis, Elias; Kribs, David

doi:10.1090/S0002-9947-05-03692-5

Applications of the Wold decomposition to the study of row contractions associated with directed graphs
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by Elias Katsoulis and David W. Kribs PDF

Trans. Amer. Math. Soc. 357 (2005), 3739-3755 Request permission

Abstract:

Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann’s inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn’s theorem for the $\mathrm {C}^*$-algebra of a CKT family, and prove a universality theorem for $\mathrm {C}^*$-algebras generated by these families. In both cases, the $\mathrm {C}^*$-algebras generated by quiver algebras play the universal role.

References

T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 155193
William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
C. Badea and G. Cassier, Constrained von Neumann inequalities, Adv. Math. 166 (2002), no. 2, 260–297. MR 1895563, DOI 10.1006/aima.2001.2035
Teresa Bates, Jeong Hee Hong, Iain Raeburn, and Wojciech Szymański, The ideal structure of the $C^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176. MR 1988256
J.A. Ball, V. Vinnikov, Functional models for representations of the Cuntz algebra, preprint, 2002.
Marek Bożejko, Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality, Studia Math. 95 (1989), no. 2, 107–118. MR 1038498, DOI 10.4064/sm-95-2-107-118
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
L. A. Coburn, The $C^{\ast }$-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722–726. MR 213906, DOI 10.1090/S0002-9904-1967-11845-7
M. J. Crabb and A. M. Davie, von Neumann’s inequality for Hilbert space operators, Bull. London Math. Soc. 7 (1975), 49–50. MR 365179, DOI 10.1112/blms/7.1.49
David Crocker, Alexander Kumjian, Iain Raeburn, and Dana P. Williams, An equivariant Brauer group and actions of groups on $C^*$-algebras, J. Funct. Anal. 146 (1997), no. 1, 151–184. MR 1446378, DOI 10.1006/jfan.1996.3010
Kenneth R. Davidson, Elias Katsoulis, and David R. Pitts, The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99–125. MR 1823866, DOI 10.1515/crll.2001.028
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
Neal J. Fowler and Iain Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), no. 1, 155–181. MR 1722197, DOI 10.1512/iumj.1999.48.1639
Neal J. Fowler and Iain Raeburn, Discrete product systems and twisted crossed products by semigroups, J. Funct. Anal. 155 (1998), no. 1, 171–204. MR 1623138, DOI 10.1006/jfan.1997.3227
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
I. Hirshberg, On the universal property of Pimsner-Toeplitz $\mathrm {C}^*$-algebras and their continuous analogues, e-print arxiv math.OA/0310407, preprint, 2003
John A. Holbrook, Schur norms and the multivariate von Neumann inequality, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 375–386. MR 1902811
F. Jaeck, S.C. Power, Hyper-reflexivity of free semigroupoid algebras, e-print arxiv math.OA/0311201, preprint, 2003.
Michael T. Jury and David W. Kribs, Partially isometric dilations of noncommuting $N$-tuples of operators, Proc. Amer. Math. Soc. 133 (2005), no. 1, 213–222. MR 2085172, DOI 10.1090/S0002-9939-04-07547-1
M.T. Jury, D.W. Kribs, Ideal structure in free semigroupoid algebras from directed graphs, J. Operator Theory, to appear.
E. Katsoulis, D.W. Kribs, The $C^*$-envelope of the tensor algebra of a directed graph, preprint, 2004.
Elias Katsoulis and David W. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann. 330 (2004), no. 4, 709–728. MR 2102309, DOI 10.1007/s00208-004-0566-6
Alex Kumjian and David Pask, $C^*$-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1503–1519. MR 1738948, DOI 10.1017/S0143385799151940
Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, DOI 10.2140/pjm.1998.184.161
Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. MR 1432596, DOI 10.1006/jfan.1996.3001
David W. Kribs and Stephen C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2004), no. 2, 117–159. MR 2076898
D.W. Kribs, S.C. Power, Partly free algebras, Proc. International Workshop on Operator Theory and its Applications, Operator Theory: Advances and Applications, Birkhäuser-Verlag Basel/Switzerland, 149 (2004), 381-393.
Donald E. Marshall, Blaschke products generate $H^{\infty }$, Bull. Amer. Math. Soc. 82 (1976), no. 3, 494–496. MR 402054, DOI 10.1090/S0002-9904-1976-14071-2
Paul S. Muhly, A finite-dimensional introduction to operator algebra, Operator algebras and applications (Samos, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 495, Kluwer Acad. Publ., Dordrecht, 1997, pp. 313–354. MR 1462686
Paul S. Muhly and Baruch Solel, Tensor algebras, induced representations, and the Wold decomposition, Canad. J. Math. 51 (1999), no. 4, 850–880. MR 1701345, DOI 10.4153/CJM-1999-037-8
Paul S. Muhly and Baruch Solel, Tensor algebras over $C^*$-correspondences: representations, dilations, and $C^*$-envelopes, J. Funct. Anal. 158 (1998), no. 2, 389–457. MR 1648483, DOI 10.1006/jfan.1998.3294
Michael V. Pimsner, A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\textbf {Z}$, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212. MR 1426840
A. I. Vinogradov, Artin’s $L$-series and the adele group, Trudy Mat. Inst. Steklov. 112 (1971), 105–122, 387 (Russian). Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, I. MR 0342494
Gelu Popescu, von Neumann inequality for $(B({\scr H})^n)_1$, Math. Scand. 68 (1991), no. 2, 292–304. MR 1129595, DOI 10.7146/math.scand.a-12363
Gelu Popescu, Multi-analytic operators and some factorization theorems, Indiana Univ. Math. J. 38 (1989), no. 3, 693–710. MR 1017331, DOI 10.1512/iumj.1989.38.38033
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
Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR 822228
Baruch Solel, You can see the arrows in a quiver operator algebra, J. Aust. Math. Soc. 77 (2004), no. 1, 111–122. MR 2069028, DOI 10.1017/S1446788700010181
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
Wojciech Szymański, General Cuntz-Krieger uniqueness theorem, Internat. J. Math. 13 (2002), no. 5, 549–555. MR 1914564, DOI 10.1142/S0129167X0200137X
N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100. MR 0355642, DOI 10.1016/0022-1236(74)90071-8
Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124