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Toeplitz $$C^*$$-algebras on bounded symmetric domains. (English) Zbl 0549.46031
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
##### Citations:
Zbl 0408.47031; Zbl 0436.47021
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