The symbols of the generating Toeplitz operators are chosen to be suitable extensions to he MathML source">B2 of families he MathML source">S of bounded functions on he MathML source">D. Symbol classes he MathML source">S that generate important classical commutative and non-commutative Toeplitz algebras in he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si38.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=f75ca6f234696a0b92870e2dc47bd1f5">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 he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si9.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=e5f13752e05d1a391135e630db690b9a">
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 he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si10.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=746bc3f6164d8c5d53097bf83fed887a">
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 he MathML source">C⁎-algebras. Their Calkin algebras are described and index formulas are provided.