Abstract
In these notes we describe the recent progress made in the study of multivariable Toeplitz operators on domains in ℂn, the C*-algebras generated by these operators and the index theory associated with C*-algebra extensions of Toeplitz type. These results are important for a better understanding of multivariable complex analysis and also connect Toeplitz operators with interesting C*-algebras not of type I, namely foliation C*-algebras and irrational rotation algebras.
Supported by NSF-Grant 8702371
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References
S. Axler, J. Conway and G. McDonald, Toephiz operators on Bergman spaces, Can. J. Math. 34 (1982), 466–482.
P. Baum, R.G. Douglas, M.E. Taylor, Cycles and relative cycles in analytic K-homology, J. Diff. Geom. (to appear).
C.A. Berger, L.A. Coburn, A. Korányi, Opérateurs de Wiener-Hopf sur les sphè res de Lie, C.R. Acad. Sci. Paris Sér. A-B 290 (1980), 989–991.
B. Blackadar, K-Theory for Operator Algebras, New York, Springer, 1986.
L. Boutet de Monvel, On the index of Toeplitz operators of several complex variables, Inventiones Math. 50 (1979), 249–272.
L.A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973), 433–439.
A. Connes, A survey of foliations and operator algebras. In: Operator Algebras and Applications (R.V. Kadison, ed.), Proc. Symp. Pure Math. 38, Amer. Math. Soc., Providence, R.I., 1981.
R.E. Curto, P.S. Muhly, C*-algebras of multiplication operators on Bergman spaces, J. Funct. Anal. 64 (1985), 315–329.
R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
R.G. Douglas, S. Hurder, J. Kaminker, Toeplitz operators and the Eta invariant.The case of S1, preprint.
A. Dynin, Inversion problem for singular integral operators: C*-approach, Proc. Natl. Acad. Sci. USA 75 (1978), 4668–4670.
D. Handelman, H.-S. Yin, Toeplitz algebras and rotational automorphisms associated to polydiscs, Amer. J. Math, (to appear).
S. Krantz, Function Theory of Several Complex Variables, New York, Wiley, 1982.
O. Loos, Bounded Symmetric Domains and Jordan Pairs, Univ. of California, Irvine, 1977.
P.S. Muhly, J.N. Renault, C*-algebras of multi-variable Wiener-Hopf operators, Trans. Amer. Math. Soc. 274 (1983), 1–44.
I. Raeburn, On Toeplitz operators associated with strongly pseudoconvex domains, Studia Math. 63 (1979), 253–258.
J. Renault, A Groupoid Approach to C*-Algebras, Lect. Notes in Math. 793, New York, Springer, 1980.
N. Salinas, A. Sheu, H. Upmeier, Toeplitz operators on pseudoconvex domains and foliation C*-algebras, preprint.
H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221–237.
H. Upmeier, Toeplitz C*-algebras on bounded symmetric domains, Ann. Math. 119 (1984), 549–576.
H. Upmeier, Fredholm indices for Toeplitz operators on bounded symmetric domains, Amer. J. Math, (to appear).
U. Venugopalkrishna, Fredholm operators associated with strongly pseudoconvex domains, J. Funct. Anal. 9 (1972), 349–373.
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Upmeier, H. (1991). Multivariable Toeplitz Operators and Index Theory. In: Araki, H., Kadison, R.V. (eds) Mappings of Operator Algebras. Progress in Mathematics, vol 84. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0453-4_16
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DOI: https://doi.org/10.1007/978-1-4612-0453-4_16
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