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Tensor Algebras, Induced Representations, and the Wold Decomposition

Published online by Cambridge University Press:  20 November 2018

Paul S. Muhly  and
Baruch Solel
Paul S. Muhly
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A. email: [email protected]
Baruch Solel
Affiliation:
Department of Mathematics, Technion, 32000 Haifa, Israel email: [email protected]
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Abstract

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Our objective in this sequel to $[18]$ is to develop extensions, to representations of tensor algebras over ${{C}^{*}}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling’s theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

Keywords

46L0546L4046L8947D1547D2546M1046M9947A2047A4547B35Tensor AlgebrasCorrespondenceInduced RepresentationWold DecompositionBeurling’s theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 4 , 01 August 1999 , pp. 850 - 880
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Wm. Arveson, Subalgebras of C*-algebras. ActaMath. 123(1969), 141–224.Google Scholar
[2] Beurling, A., On two problems concerning linear transformations in Hilbert space. ActaMath. 81(1949), 239–255.Google Scholar
[3] Blecher, D., Ruan, Z.-J., and Sinclair, A., A characterization of operator algebras. J. Funct. Anal. 89(1990), 188–201.Google Scholar
[4] Davidson, K. and Pitts, D., The algebraic structue of non-commutative analytic Toeplitz algebras. Math. Ann. 311(1998), 275–303.Google Scholar
[5] Davidson, K. and Pitts, D., Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc., to appear.Google Scholar
[6] Coburn, L., The C*-algebra generated by an isometry. Bull. Amer.Math. Soc. 73(1967), 722–726.Google Scholar
[7] Dixmier, J., C*-Algebras. North-Holland, 1977.Google Scholar
[8] Doplicher, S., Pinzari, C., and Zuccante, R., The C*-algebra of a Hilbert bimodule. Boll. Un.Mat. Ital. B (8) 1(1998), 263–281.Google Scholar
[9] Doplicher, S. and Roberts, J., A new duality theory for compact groups. Invent.Math. 98(1989), 157–218.Google Scholar
[10] Douglas, R. G. and Paulsen, V., Hilbert Modules for Function Algebras. Research Notes in Math. 217, Longman, 1989.Google Scholar
[11] Fowler, N. and Raeburn, I., The Toeplitz algebra of a Hilbert bimodule. Preprint.Google Scholar
[12] Halmos, P., Shifts on Hilbert spaces. J. Reine Angew.Math. 208(1961), 102–112.Google Scholar
[13] Jensen, K. K. and Thomsen, K., Elements of KK-Theory. Birkäuser, Boston, Bassel, Berlin, 1991.Google Scholar
[14] Lax, P., Translation invariant subspaces. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, 299–306.Google Scholar
[15] Muhly, P. and Solel, B., Hilbert modules over operator algebras. Mem. Amer.Math. Soc. (559) 117(1995).Google Scholar
[16] Muhly, P. and Solel, B., On the uniqueness of operator algebra structures. Indiana Univ.Math. J. 46(1997), 575–591.Google Scholar
[17] Muhly, P. and Solel, B., An algebraic characterization of boundary representations. In: Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics, Advances in Operator Theory 104, Birkhäuser, Basel, 1998, 189–196.Google Scholar
[18] Muhly, P. and Solel, B., Tensor algebras over C*-correspondences (Representations, dilations, and C*-envelopes). J. Funct. Anal. 158(1998), 389–457.Google Scholar
[19] Pimsner, M., A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed products by Z. In: Free Probability Theory, (ed. Voiculescu, D.), Fields Institute Communications 12, Amer.Math. Soc., Providence, 1997, 189–212.Google Scholar
[20] Popescu, G., Isometric dilations for infinite sequences of noncommuting operators. Trans. Amer. Math. Soc. 316(1989), 523–536.Google Scholar
[21] Popescu, G., Characteristic functions for infinite sequences of noncommuting operators. J. Operator Theory 22(1989), 51–71.Google Scholar
[22] Popescu, G., Multi-analytic operators on Fock spaces. Math. Ann. 303(1995), 31–46.Google Scholar
[23] Popescu, G., NoncommutativeWold decompositions for semigroups of isometries. IndianaUniv.Math. J. 47(1998), 277–296.Google Scholar
[24] Popescu, G., A generalization of Beurling's theorem to a class of reflexive algebras. J. Operator Theory, to appear.Google Scholar
[25] Rieffel, M., Induced representations of C*-algebras. Adv. inMath. 13(1974), 176–257.Google Scholar
[26] Rosenblum, M., Vectorial Toeplitz operators and the Fejer-Riesz theorem. J. Math.Anal. Appl. 23(1968), 139–147.Google Scholar