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Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains

Author: Norberto Salinas
Journal: Trans. Amer. Math. Soc. 336 (1993), 675-699
MSC: Primary 47B35; Secondary 32A07, 46L05, 47B38
MathSciNet review: 1093217
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Abstract: The notion of weighted Wiener-Hopf operators is introduced. Their relationship with Toeplitz operators acting on the space of holomorphic functions which are square integrable with respect to a given "symmetric" measure is discussed. The groupoid approach is used in order to present a general program for studying the $ {C^{\ast} }$-algebra generated by weighted Wiener-Hopf operators associated with a solid cone of a second countable locally compact Hausdorff group. This is applied to the case when the group is the dual of a connected locally compact abelian Lie group and the measure is "well behaved" in order to produce a geometric groupoid which is independent of the representation. The notion of a Reinhardt-tube domain $ \Omega $ appears thus naturally, and a decomposition series of the corresponding $ {C^{\ast} }$-algebra is presented in terms of groupoid $ {C^{\ast} }$-algebras associated with various parts of the boundary of the domain $ \Omega $.

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