Noncommutative complex analysis and Bargmann-Segal multipliers
HTML articles powered by AMS MathViewer
- by Richard Rochberg and Nik Weaver PDF
- Proc. Amer. Math. Soc. 129 (2001), 2679-2687 Request permission
Abstract:
We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on $L^{2}(\mathbf {R})$ which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro, which act as unbounded multiplication operators on Bargmann-Segal space.References
- J. Aniansson, Some Integral Representations in Real and Complex Analysis, Doctoral Thesis, Royal Institute of Technology, Stockholm (1999).
- S. Twareque Ali, J.-P. Antoine, J.-P. Gazeau, and U. A. Mueller, Coherent states and their generalizations: a mathematical overview, Rev. Math. Phys. 7 (1995), no. 7, 1013–1104. MR 1359988, DOI 10.1142/S0129055X95000396
- C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 813–829. MR 882716, DOI 10.1090/S0002-9947-1987-0882716-4
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- A. A. Borichev, The polynomial approximation property in Fock-type spaces, Math. Scand. 82 (1998), 256-264.
- Dariusz Cichoń and Jan Stochel, On Toeplitz operators in Segal-Bargmann spaces, Univ. Iagel. Acta Math. 34 (1997), 35–43. MR 1458030
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- Edward G. Effros, Advances in quantized functional analysis, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 906–916. MR 934293
- Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
- J. Janas, Unbounded Toeplitz operators in the Bargmann-Segal space, Studia Math. 99 (1991), no. 2, 87–99. MR 1120741, DOI 10.4064/sm-99-2-87-99
- Jan Janas and Jan Stochel, Unbounded Toeplitz operators in the Segal-Bargmann space. II, J. Funct. Anal. 126 (1994), no. 2, 418–447. MR 1305075, DOI 10.1006/jfan.1994.1153
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- J. von Neumann, Die eindeutigkeit der Schrödingerschen operatoren, Math. Ann. 104 (1931), 570-578.
- D. J. Newman and H. S. Shapiro, Fischer spaces of entire functions, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 360–369. MR 0234012
- —A Hilbert space of entire functions related to the operational calculus, manuscript.
- Marc A. Rieffel, Noncommutative tori—a case study of noncommutative differentiable manifolds, Geometric and topological invariants of elliptic operators (Brunswick, ME, 1988) Contemp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 191–211. MR 1047281, DOI 10.1090/conm/105/1047281
- Kristian Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I, J. Reine Angew. Math. 429 (1992), 91–106. MR 1173117, DOI 10.1515/crll.1992.429.91
- S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), no. 1, 125–170. MR 994499, DOI 10.1007/BF01221411
Additional Information
- Richard Rochberg
- Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
- MR Author ID: 149315
- Email: rr@math.wustl.edu
- Nik Weaver
- Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
- MR Author ID: 311094
- Email: nweaver@math.wustl.edu
- Received by editor(s): September 27, 1999
- Received by editor(s) in revised form: January 14, 2000
- Published electronically: February 9, 2001
- Communicated by: David R. Larson
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2679-2687
- MSC (2000): Primary 46L89, 47B32; Secondary 30D15
- DOI: https://doi.org/10.1090/S0002-9939-01-05897-X
- MathSciNet review: 1838792