Toeplitz \(C^*\)-algebras on bounded symmetric domains. (English) Zbl 0549.46031

Upmeier, Harald

Ann. Math. (2) 119, 549-576 (1984).

This paper gives a structure theory for the \(C^*\)-algebra \({\mathcal T}\) generated by all Toeplitz operators \(T_ f\) with continuous symbol function \(f\in C(S)\), acting on the Hardy space \(H^ 2(S)\) associated with the Shilov boundary S of a bounded symmetric domain D in \({\mathbb{C}}^ n\) of arbitrary rank r. The main result shows that \({\mathcal T}\) is a solvable \(C^*\)-algebra of length r [in the sense of A. Dynin, Proc. Natl. Acad. Sci. USA 75, 4668-4670 (1978; Zbl 0408.47031)]. Its subquotients can be described in terms of the Jordan algebraic structure underlying the domain D. Domains of rank 1 and tube domains of rank 2 have already been studied by C. A. Berger, L. A. Coburn and A. Korányi [C. R. Acad. Sci., Paris, Sér. A 290, 989-991 (1980; Zbl 0436.47021)].

MSC:

46L05	General theory of \(C^*\)-algebras
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators

Keywords:

Toeplitz \(C^*\)-algebras; bounded symmetric domains; Jordan algebras; Toeplitz operators; continuous symbol; Hardy space; Shilov boundary; solvable \(C^*\)-algebra; Jordan algebraic structure

Citations:

Zbl 0408.47031; Zbl 0436.47021