This talk aims to extend the investigation of C*-algebras generated by con-

tinuous Toeplitz operators and linear fractional composition operators on the Hardy space

to certain classes of discontinuous Toeplitz symbols. We show that the C*-algebra generated

by quasicontinuous Toeplitz operators and a group of automorphic composition operators is

isomorphic modulo the compact operators to the crossed product of the C*-algebra of qua-

sicontinuous functions by the group acting as compositions of these functions. Specializing

to the case of a single rational rotation generator, we characterize the Fredholm operators

in the algebra and discuss their indices. These results extend certain results of M. Jury

(2007). Next, we consider the commutative non-self-adjoint Calkin subalgebra generated

by the composition operator of a non-parabolic linear fractional non-automorphism xing a

boundary point and a certain class of piecewise quasicontinuous Toeplitz operators. Its max-

imal ideal space and Shilov boundary are identied, which yields essential spectral bounds

upon spectral permanence considerations in the Calkin algebra. These results are motivated

by Quertermous (2013) which obtained an isomorphism of a noncommutative C*-algebra.