The symbols of the generating Toeplitz operators are chosen to be suitable extensions to 82a8ed90c134c8d437fec2b5699" title="Click to view the MathML source">B2 of families e67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">S of bounded functions on 8aa1939c159e2717739dde9fbd" title="Click to view the MathML source">D. Symbol classes e67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">S that generate important classical commutative and non-commutative Toeplitz algebras in bd1f5"> are of particular interest. In this paper we discuss various examples. In the case of e630db690b9a"> and 83fed887a"> we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.
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