The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in mage" height="21" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si38.gif"> are of particular interest. In this paper we discuss various examples. In the case of
mage" height="18" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si9.gif"> and
mage" height="18" width="151" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si10.gif"> we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.