zbMATH — the first resource for mathematics

Toeplitz operators on Jordan-Kepler varieties. (English) Zbl 1447.47033
Summary: We study Toeplitz operators on Hilbert spaces of holomorphic functions on algebraic varieties, the generalized Kepler varieties defined in Jordan theoretic terms. Using the fine analysis of the reproducing kernel function, we construct and classify the irreducible representations of the \(C^{*}\)-algebra generated by these operators (with smooth symbols). The limit behavior of Bergman-type measures under peaking functions is of special importance.

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
47C15 Linear operators in \(C^*\)- or von Neumann algebras
Full Text: DOI Euclid
[1] J. Arazy, M. Engliš, and W. Kaup, Holomorphic retractions and boundary Berezin transforms, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 641–657. · Zbl 1176.47026
[2] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994.
[3] R. E. Curto and P. S. Muhly, \(C^{*}\)-algebras of multiplication operators on Bergman spaces, J. Funct. Anal. 64 (1985), no. 3, 315–329. · Zbl 0583.46049
[4] M. Engliš and H. Upmeier, Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds, preprint, arXiv:1708.03388v1 [math.CV].
[5] J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), no. 1, 64–89. · Zbl 0718.32026
[6] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., Oxford Univ. Press, New York, 1994.
[7] N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ. 39, Amer. Math. Soc., Providence, 1968. · Zbl 0218.17010
[8] O. Loos, Bounded Symmetric Domains and Jordan Pairs, lectures, University of California, Irvine, 1977.
[9] E. Neher, Jordan Triple Systems by the Grid Approach, Lecture Notes in Math. 1280, Springer, Berlin, 1987. · Zbl 0621.17001
[10] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970), 61–80. · Zbl 0219.32013
[11] H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280, no. 1 (1983), 221–237. · Zbl 0527.47019
[12] H. Upmeier, Toeplitz \(C^{*}\)-algebras on bounded symmetric domains, Ann. of Math. (2) 119 (1984), no. 3, 549–576. · Zbl 0549.46031
[13] H. Upmeier, Toeplitz Operators and Index Theory in Several Complex Variables, Oper. Theory Adv. Appl. 81, Birkhäuser, Basel, 1996. · Zbl 0957.47023
[14] H. Upmeier and K. Wang, Dixmier trace for Toeplitz operators on bounded symmetric domains, J. Funct. Anal. 271 (2016), no. 3, 532–565. · Zbl 1359.32018
[15] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. Lond. Math. Soc. (2) 46 (1940), 389–408. · JFM 66.1243.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.