Salinas, Norberto; Sheu, Albert; Upmeier, Harald Toeplitz operators on pseudoconvex domains and foliation \(C^*\)- algebras. (English) Zbl 0708.47021 Ann. Math. (2) 130, No. 3, 531-565 (1989). 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 Cited in 1 ReviewCited in 27 Documents 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) Keywords:Toeplitz operators over arbitrary bounded pseudoconvex domains; multivariable Toeplitz operators; strongly pseudoconvex; finite type; symmetric; Koszul complex; Toeplitz \(C^ *\)-algebra; pseudo-convex Reinhardt domains; foliation \(C^ *\)-algebras; \({\bar \partial }\)- Neumann operator on (0,1)-forms with analytic coefficients; Bergmann Toeplitz operators; irrational rotation algebras Citations:Zbl 0549.46031 PDFBibTeX XMLCite \textit{N. Salinas} et al., Ann. Math. (2) 130, No. 3, 531--565 (1989; Zbl 0708.47021) Full Text: DOI