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Toeplitz algebras on strongly pseudoconvex domains. (English) Zbl 1134.47056

It is shown in this paper that the \(K_0\)-group for the Toeplitz \(C^*\)-algebra \(\mathfrak T_{\partial \Omega}\) associated with the boundary \(\partial \Omega\) of a strongly pseudoconvex domain \(\Omega\) is isomorphic to that for the \(C^*\)-algebra \(C(\partial \Omega)\) of continuous functions on \(\partial \Omega\), which is identified with the topological \(K_0\)-group of \(\partial \Omega\), and the \(K_1\)-group of \(C(\partial \Omega)\) is the extension of the \(K_1\)-group of \(\mathfrak T_{\partial \Omega}\) by \(\mathbb Z\). Moreover, \(K\)-groups for some illustrative particular examples in these obtained formulas are computed explicitly. The first cohomology group for \(\mathfrak T_{\partial \Omega}\) is considered as well.

MSC:

47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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References:

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