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K-theory of Toeplitz \(C^*\)-algebras on Lie spheres. (English) Zbl 0731.46037
Let D be a symmetric domain in the complex plane with the Shilov boundary \(S_ D\) and let \(P: L^ 2(S_ D)\to H^ 2(S_ D)\) be the orthogonal projection onto the Hardy space. The author considers the Toeplitz operators \(T_ f\phi =P(f\phi),\phi \in H^ 2(S_ D)\), with matricial symbols f. He computes the K-groups of the \(C^*\)-algebra of these operators and obtains the theorem on Fredholmness and index of \(T_ f\). This theorem generalises the similar result for scalar-valued symbol, see C. A. Berger, L. A. Coburn and A. Coranyi, C. R. Acad. Sci., Paris, Sér. A 290, 989-991 (1980; Zbl 0436.47021).
MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47C15 Linear operators in \(C^*\)- or von Neumann algebras
46L87 Noncommutative differential geometry
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