The symbols of the generating Toeplitz operators are chosen to be suitable extensions to 8d437fec2b5699" title="Click to view the MathML source">B2 of families S of bounded functions on bd" title="Click to view the MathML source">D. Symbol classes 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 e5f13752e05d1a391135e630db690b9a"> and we characterize all irreducible representations of the resulting Toeplitz operator 8d4d35557d1cba9a54ff005" title="Click to view the MathML source">C⁎-algebras. Their Calkin algebras are described and index formulas are provided.
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