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Toeplitz operators on pseudoconvex domains and foliation \(C^*\)- algebras. (English) Zbl 0708.47021

In this far-reaching paper, the authors initiate a study of Toeplitz operators over arbitrary bounded pseudoconvex domains on \({\mathbb{C}}^ n\). Previous work on multivariable Toeplitz operators has concentrated on special classes of domains such as strongly pseudoconvex, or finite type, or symmetric. The main results of the paper include:
1. A description of the Koszul complex associated with the Toeplitz operators on an arbitrary pseudoconvex bounded domain, subject to a mild geometric condition.
2. A complete analysis of the Toeplitz \(C^*\)-algebra \({\mathcal T}(D)\) for pseudo-convex Reinhardt domains, given in terms of foliation \(C^*\)- algebras induced by the boundary geometry of the underlying domain. The result is a description of \({\mathcal T}(D)\) as a composition series, analogous to the case of symmetric domains, where the boundary components actually form a fibration [D. Upmeier, “Toeplitz \(C^*\)-algebras on bounded symmetric domains”, Ann. Math., II. Ser. 119, 549-576 (1984; Zbl 0549.46031)].
3. A geometric description of the \({\bar \partial}\)-Neumann operator on (0,1)-forms with analytic coefficients and, more generally of the type I nature of \({\mathcal T}(D)\), for D a Reinhardt domain. The authors also exhibit a large class of pseudoconvex Reinhardt domains in \({\mathbb{C}}^ 2\) for which not only the Neumann operators fails to be convex, but T(D) is not even of type I.
4. An index formula for certain Reinhardt domains of “irrational” type, relating to Bergmann Toeplitz operators and irrational rotation algebras.
Reviewer: J.Butz

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32T99 Pseudoconvex domains
46L05 General theory of \(C^*\)-algebras
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
46L55 Noncommutative dynamical systems
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)

Citations:

Zbl 0549.46031
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