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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: [email protected]
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Abstract

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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

47L3046E2247A13non-selfadjoint operator algebrasreproducing kernelHilbert spacesmultiplier algebraNevanlinna-Pick kernelsisomorphism 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

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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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 Google 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.Google 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 Google 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