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Toeplitz operators on symmetric Siegel domains. (English) Zbl 0565.47016
In this paper, the structure theory for Toeplitz \(C^*\)-algebras over bounded symmetric domains in \({\mathbb{C}}^ n\) [cf. H. Upmeier, Ann. Math., II. Ser. 119, 549-576 (1984; Zbl 0549.46031)] is applied to study smooth Toeplitz operators on the distinguished boundary \(\Sigma\) of a symmetric Siegel domain D. As the main result, it is shown that the \(C^*\)-algebra \({\mathcal T}_{\Sigma}\) generated by these operators is solvable of length \(r=rank(D)\) and that its irreducible representations correspond to the boundary components of D. This applies in particular to Wiener-Hopf operators on self-dual homogeneous cones, first studied by P. S. Muhly and J. N. Renault [Trans. Am. Math. Soc. 274, 1- 44 (1982; Zbl 0509.46049)].

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46L05 General theory of \(C^*\)-algebras
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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