Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 012, 22 pages      arXiv:1410.0733      https://doi.org/10.3842/SIGMA.2015.012

The Quantum Pair of Pants

Slawomir Klimek a, Matt Mcbride b, Sumedha Rathnayake a and Kaoru Sakai a
a) Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA
b) Department of Mathematics, University of Oklahoma, 601 Elm St., Norman, OK 73019, USA

Received October 24, 2014, in final form February 03, 2015; Published online February 10, 2015

Abstract
We compute the spectrum of the operator of multiplication by the complex coordinate in a Hilbert space of holomorphic functions on a disk with two circular holes. Additionally we determine the structure of the $C^*$-algebra generated by that operator. The algebra can be considered as the quantum pair of pants.

Key words: quantum domains; $C^*$-algebras.

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References

  1. Abrahamse M.B., Toeplitz operators in multiply connected regions, Amer. J. Math. 96 (1974), 261-297.
  2. Abrahamse M.B., Douglas R.G., Operators on multiply connected domains, Proc. Roy. Irish Acad. Sect. A 74 (1974), 135-141.
  3. Ahlfors L.V., Complex analysis. An introduction to the theory of analytic functions of one complex variable, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.
  4. Coburn L.A., Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973), 433-439.
  5. Conway J.B., A course in operator theory, Graduate Studies in Mathematics, Vol. 21, Amer. Math. Soc., Providence, RI, 2000.
  6. Halmos P.R., Sunder V.S., Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 96, Springer-Verlag, Berlin - New York, 1978.
  7. Klimek S., Lesniewski A., Quantum Riemann surfaces. I. The unit disc, Comm. Math. Phys. 146 (1992), 103-122.
  8. Klimek S., Lesniewski A., Quantum Riemann surfaces. II. The discrete series, Lett. Math. Phys. 24 (1992), 125-139.
  9. Klimek S., Lesniewski A., A two-parameter quantum deformation of the unit disc, J. Funct. Anal. 115 (1993), 1-23.
  10. Klimek S., Lesniewski A., Quantum Riemann surfaces. III. The exceptional cases, Lett. Math. Phys. 32 (1994), 45-61.
  11. Klimek S., Lesniewski A., Quantum Riemann surfaces for arbitrary Planck's constant, J. Math. Phys. 37 (1996), 2157-2165.
  12. Klimek S., McBride M., D-bar operators on quantum domains, Math. Phys. Anal. Geom. 13 (2010), 357-390, arXiv:1001.2216.
  13. Klimek S., McBride M., A note on Dirac operators on the quantum punctured disk, SIGMA 6 (2010), 056, 12 pages, arXiv:1003.5618.
  14. Klimek S., McBride M., Classical limit of the d-bar operators on quantum domains, J. Math. Phys. 52 (2011), 093501, 16 pages, arXiv:1101.2645.
  15. Klimek S., McBride M., A note on gluing Dirac type operators on a mirror quantum two-sphere, SIGMA 10 (2014), 036, 15 pages, arXiv:1309.7096.
  16. Markushevich A.I., Theory of functions of a complex variable, Chelsea Publishing Co., New York, 2005.

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