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On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces

Published online by Cambridge University Press:  20 November 2018

Michael Hartz
Michael Hartz
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 e-mail: mphartz@uwaterloo.ca
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E-mail address:
mphartz@uwaterloo.ca
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Abstract

We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.

Keywords

47L30 46E22 47A13 non-selfadjoint operator algebras reproducing kernelHilbert spaces multiplier algebra Nevanlinna-Pick kernels isomorphism problem.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 1 , 01 February 2017 , pp. 54 - 106
Copyright
Copyright © Canadian Mathematical Society 2017

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References

[1] Agler, J. and McCarthy, J. E., Complete Nevanlinna-Pick kernels. J. Funct. Anal. 175(2000), no. 1, 111–124. http://dx.doi.Org/10.1006/jfan.2000.3599 CrossRefGoogle Scholar
[2] Agler, J., Nevanlinna-Pick kernels and localization. In: Operator theoretical methods (Timisoara,1998), Tneta Found., Bucharest, 2000, pp. 1–20.Google Scholar
[3] Agler, J., Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, 44, American Mathematical Society, Providence, RI, 2002.Google Scholar
[4] Alpay, D., Putinar, M., and Vinnikov, V. , A Hilbert space approach to bounded analytic extension in the ball. Commun. Pure Appl. Anal. 2(2003), no. 2,139–145. http://dx.doi.Org/10.3934/cpaa.2003.2.139 Google Scholar
[5] Arcozzi, N., Rochberg, R., and Sawyer, E., Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls. Adv. Math. 218(2008), no. 4,1107–1180.http://dx.doi.Org/10.1016/j.aim.2008.03.001 CrossRefGoogle Scholar
[6] Aronszajn, N., Theory of reproducing kernels. Trans. Amer. Math. Soc. 68(1950), 337–404.http://dx.doi.org/10.1090/S0002-9947-1950-00514377 CrossRefGoogle Scholar
[7] Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.Google Scholar
[8] Cowen, C. C., Some problems on composition operators, Studies on composition operators (Laramie, WY, 1996). Contemp. Math., 213, American Mathematical Society, Providence, RI, 1998, pp. 17–25. http://dx.doi.org/10.1090/conm/213/02846, Google Scholar
[9] Curto, R. E., Fredholm and invertible n-tuples of operators. The deformation problem. Trans. Amer. Math. Soc. 266(1981), no. 1, 129–159.http://dx.doi.Org/10.2307/1998391 Google Scholar
[10] Dales, H. G., Automatic continuity: a survey. Bull. London Math. Soc. 10(1978), no. 2,129–183. http://dx.doi.Org/10.1112/blms/10.2.129 CrossRefGoogle Scholar
[11] Davidson, K. R., Hartz, M., and Shalit, O. M. , Erratum to: Multipliers of embedded discs. Complex Analysis and Operator Theory 9(2015), no. 2, 323–327. http://dx.doi.Org/10.1007/s11785-014-0400-4 CrossRefGoogle Scholar
[12] Davidson, K. R., Multipliers of embedded discs. Complex Analysis and Operator Theory 9(2015), no 2, 287-321; Erratum, ibid., 323–327. http://dx.doi.Org/10.1007/s11785-014-0360-8 CrossRefGoogle Scholar
[13] Davidson, K. R. and Pitts, D. R., The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311(1998), no. 2, 275-303; Erratum, Math. Ann. 361 (2015), no. 3-4, 1123–1124. http://dx.doi.Org/10.1007/S002080050188 CrossRefGoogle Scholar
[14] Davidson, K. R.,Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. Integral Equations Operator Theory 31(1998), no. 3, 321–337. http://dx.doi.Org/10.1007/BF01195123 CrossRefGoogle Scholar
[15] Davidson, K. R., Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. (3) 78(1999), no. 2, 401–430. http://dx.doi.Org/10.1112/S002461159900180X CrossRefGoogle Scholar
[16] Davidson, K. R., Ramsey, C., and Moshe Shalit, O., The isomorphism problem for some universal operator algebras. Adv. Math. 228 (2011), no. 1,167–218.http://dx.doi.Org/10.1016/j.aim.2O11.05.015 CrossRefGoogle Scholar
[17] Davidson, K. R.,Operator algebras for analytic varieties. Trans. Amer. Math. Soc. 367(2015), no. 2, 1121–1150. http://dx.doi.org/10.1090/S0002-9947-2014-05888-1 CrossRefGoogle Scholar
[18] Eschmeier, J. and Putinar, M., Spectral decompositions and analytic sheaves. London Mathematical Society Monographs. New Series, 10, The Clarendon Press, Oxford University Press, New York, 1996.Google Scholar
[19] Feller, W., An introduction to probability theory and its applications. Vol. I, Third ed., John Wiley & Sons Inc., New York, 1968.Google Scholar
[20] Garnett, J. B., Bounded analytic functions. First ed., Graduate Texts in Mathematics, 236, Springer, New York, 2007.Google Scholar
[21] Gleason, J., Richter, S., and Sundberg, C., On the index of invariant subspaces in spaces of analytic functions of several complex variables. J. Reine Angew. Math. 587(2005), 49–76. http://dx.doi.Org/10.1515/CI-II.2OO5.2005.587.49 Google Scholar
[22] Greene, D. C. V., Richter, S., and Sundberg, C., The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels. J. Funct. Anal. 194 (2002), no. 2, 311–331.http://dx.doi.org/10.1006/jfan.2002.3928 CrossRefGoogle Scholar
[23] Guo, K., Hu, J., and Xu, X., Toeplitz algebras, subnormal tuples and rigidity on reproducing C[z1, …,zd]-modules. J. Funct. Anal. 210 (2004), no. 1, 214–247.http://dx.doi.Org/10.1016/j.jfa.2003.06.003 CrossRefGoogle Scholar
[24] Hartz, M., Topological isomorphisms for some universal operator algebras. J. Funct. Anal. 263 (2012), no. 11, 3564–3587.http://dx.doi.Org/10.1016/j.jfa.2012.08.028 CrossRefGoogle Scholar
[25] Hartz, M.,Universal operator algebras for commuting row contractions. Master's thesis, Universität des Saarlandes, 2012.Google Scholar
[26] Hayden, T. L. and Suffridge, T. J., Biholomorphic maps in Hilbert space have a fixed point. Pacific J. Math. 38(1971), 419–422.CrossRefGoogle Scholar
[27] Kerr, M., McCarthy, J. E., and Shalit, O., On the isomorphism question for complete pick multiplier algebras. Integral Equations Operator Theory 76(2013), no. 1, 39–53.http://dx.doi.Org/10.1007/s00020-013-2048-2 CrossRefGoogle Scholar
[28] McCarthy, J. E. and Shalit, O. M., Spaces ofdirichlet series with the complete pick property. http://arxiv:1507.04162Google Scholar
[29] Millier, V., Spectral theory of linear operators and spectral systems in Banach algebras. Second ed., Operator Theory: Advances and Applications, 139, Birkhäuser Verlag, Basel, 2007.Google Scholar
[30] Rudin, W., Function theory in the unit ball of C. Reprint of the 1980 edition., Classics in Mathematics, Springer-Verlag, Berlin, 2008.Google Scholar
[31] Salomon, G. and Shalit, O., The isomorphism problem for complete pick algebras: a survey. arxiv:1412.7817Google Scholar
[32] Shalit, O. M. and Solel, B., Subproduct systems. Doc. Math. 14(2009), 801–868.Google Scholar
[33] Shields, A. L., Weighted shift operators and analytic function theory. In: Topics in operator theory, Math. Surveys, 13, American Mathematical Society, Providence, RI, 1974, pp. 49–128.Google Scholar
[34] Whittaker, E. T. and Watson, G. N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Fourth ed., Reprinted, Cambridge University Press, New York, 1952.Google Scholar
[35] Zariski, O. and Samuel, P., Commutative algebra. Vol. II, Reprint of the 1960 ed., Graduate Texts in Mathematics, 29, Springer-Verlag, New York-Heidelberg, 1975.Google Scholar

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