Abstract
We establish a representation theorem for Toeplitz operators on the Segal-Bargmann (Fock) space ofC n whose “symbols” have uniform radial limits. As an application of this result, we show that Toeplitz algebras on the open ball inC n are “strict deformation quantizations”, in the sense of M. Rieffel, of the continuous functions on the corresponding closed ball.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[B] Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure and Appl. Math.14, 187–214 (1961)
[BC1] Berger, C.A., Coburn, L.A.: Toeplitz operators and quantum mechanics. J. Funct. Anal.68, 273–299 (1986)
[BC2] Berger, C.A., Coburn, L.A.: Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc.301, 813–829 (1987)
[BC3] Berger, C.A., Coburn, L.A.: Heat flow and Berezin-Toeplitz estimates. Am. J. Math.116, 563–590 (1994)
[BDF] Brown, L.G., Douglas, R.G., Fillmore, P.A.: Extensions ofC *-algebras andK-homology. Ann. Math.105, 265–324 (1977)
[BLU] Borthwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization of Cartan domains. J. Funct. Anal.113, 153–176 (1993)
[C1] Coburn, L.A.: Singular integral operators and Toeplitz operators on odd spheres. Indiana Univ. Math. J.23, 433–439 (1973)
[C2] Coburn, L.A.: Deformation estimates for the Berezin-Toeplitz quantization. Commun. Math. Phys.149, 415–424 (1992)
[F] Folland, G.B.: Harmonic analysis in phase space. Annals of Math. Studies. Princeton NJ, Princeton Univ. Press, 1989
[G] Guillemin, V.: Toeplitz operators inn-dimensions. Int. Eq. and Op. Thy.7, 145–205 (1984)
[H] Howe, R.: Quantum mechanics and partial differential equations. J. Funct. Anal.38, 188–254 (1980)
[KL] Klimek, S., Lesniewski, A.: Quantum Riemann surfaces I, The unit disc. Commun. Math. Phys.146, 103–122 (1992)
[R] Rieffel, M.A.: Deformation quantization for actions ofR d Memoirs of the Am. Math. Soc.106, No. 506 (1993)
[S] Sheu, A.J.: Quantization of PoissonSU(2) and its Poisson homogeneous space-the 2-sphere. Commun. Math. Phys.135, 217–232 (1991)
[V] Venugopalkrishna, U.: Fredholm operators associated with strongly pseudoconvex domains inC n. J. Funct. Anal.9, 349–373 (1972)
[WW] Whittaker, E.T., Watson, G.N.: Modern Analysis. London: Cambridge Univ. Press, 1940
[X] Xia, J.: Geometric invariants of the quantum Hall effect. Commun. Math. Phys.119, 29–50 (1988)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Coburn, L.A., Xia, J. Toeplitz algebras and Rieffel deformations. Commun.Math. Phys. 168, 23–38 (1995). https://doi.org/10.1007/BF02099582
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099582