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Toeplitz $$C^*$$-algebras and several complex variables. (English) Zbl 0833.47019
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 339-346 (1995).
The author presents a brief but deep survey of the $$C^*$$-algebraic properties of Toeplitz operators on a domain $$\Omega \subseteq \mathbb{C}^n$$. The main object of study is the $$C^*$$-algebra $${\mathcal I} (\partial \Omega)$$ generated by Toeplitz operators on a canonical Hilbert space of holomorphic functions on $$\Omega$$, with symbols in a natural function algebra over $$\Omega$$.
The main problems which are discussed here are the following:
When does the commutator ideal $${\mathcal C}$$ in $${\mathcal I} (\partial \Omega)$$ coincide with the ideal of all compact operators $${\mathcal K}$$?
What is the structure of $${\mathcal C}/K$$?
What is its relationship with $$\partial \Omega$$?
Theorems which give answers to these questions for strongly pseudoconvex domains are formulated and the structure of the commutator ideal $${\mathcal C}$$ is described for pseudoconvex Reinhart domains in $$\mathbb{C}^n$$ and more general $$K$$-circular domains where $$K$$ is a real compact Lie group.
For the entire collection see [Zbl 0819.00022].
##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 46L05 General theory of $$C^*$$-algebras 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 32T99 Pseudoconvex domains