The symbols of the generating Toeplitz operators are chosen to be suitable extensions to md5=4143882a8ed90c134c8d437fec2b5699" title="Click to view the MathML source">B2 of families md5=7de67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">S of bounded functions on md5=8194308aa1939c159e2717739dde9fbd" title="Click to view the MathML source">D. Symbol classes md5=7de67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">S that generate important classical commutative and non-commutative Toeplitz algebras in md5=f75ca6f234696a0b92870e2dc47bd1f5"> are of particular interest. In this paper we discuss various examples. In the case of md5=e5f13752e05d1a391135e630db690b9a"> and md5=746bc3f6164d8c5d53097bf83fed887a"> we characterize all irreducible representations of the resulting Toeplitz operator md5=bc2d4370a8d4d35557d1cba9a54ff005" title="Click to view the MathML source">C⁎-algebras. Their Calkin algebras are described and index formulas are provided.