Abstract
We prove the existence of commutative C*-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space \({{\mathbb{P}^n}\mathbb{(C)}}\). The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of \({{\mathbb{P}^n}\mathbb{(C)}}\).
Similar content being viewed by others
References
Griffiths P., Harris J.: Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. Wiley, New York (1994)
Grudsky S., Quiroga-Barranco R., Vasilevski N.: Commutative C-algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal. 234(1), 1–44 (2006)
Helgason S.: Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, vol. 80. Academic Press, Inc., New York (1978)
Knapp A.W.: Lie groups beyond an introduction, 2nd edn. Progress in Mathematics, vol. 140. Birkhuser Boston, Inc., Boston (2002)
Kobayashi S., Nomizu K.: Foundations of differential geometry, vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley, New York (1996)
Naitoh H., Takeuchi M.: Totally real submanifolds and symmetric bounded domains. Osaka J. Math. 19(4), 717–731 (1982)
Prieto Sanabria E.: Toeplitz operators on the 2-sphere. Rev. Colombiana Mat. 43(2), 87–100 (2009)
Quiroga-Barranco R., Vasilevski N.: Commutative C*-algebras of Toeplitz operators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equ. Oper. Theory 59(3), 379–419 (2007)
Quiroga-Barranco R., Vasilevski N.: Commutative C*-algebras of Toeplitz operators on the unit ball. II. Geometry of the level sets of symbols. Integral Equ. Oper. Theory 60(1), 89–132 (2008)
Quiroga-Barranco R., Vasilevski N.: Commutative algebras of Toeplitz operators on the Reinhardt domains. Integral Equ. Oper. Theory 59(1), 67–98 (2007)
Schlichenmaier, M.: Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results. Adv. Math. Phys. Art. ID 927280 (2010)
Vasilevski, N.: Toeplitz operators on the Bergman spaces: inside-the-domain effects. Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), 79146, Contemp. Math. 289, Am. Math. Soc., Providence (2001)
Vasilevski, N.: Commutative algebras of Toeplitz operators on the Bergman space. Operator theory: advances and applications, vol. 185. Birkhäuser Verlag, Basel (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
R. Quiroga-Barranco was partially supported by SNI-Mexico and by the Conacyt grant no. 82979. A. Sanchez-Nungaray was partially supported by a Conacyt postdoctoral fellowship.
Rights and permissions
About this article
Cite this article
Quiroga-Barranco, R., Sanchez-Nungaray, A. Commutative C*-Algebras of Toeplitz Operators on Complex Projective Spaces. Integr. Equ. Oper. Theory 71, 225–243 (2011). https://doi.org/10.1007/s00020-011-1897-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-011-1897-9