On the algebras generated by Toeplitz operators with piecewise continuous symbols

We explain an apparent disagreement between the fact that the Fredholm symbol algebras of two different C∗-algebras generated by Toeplitz operators with piecewise continuous symbols, acting on the Hardy space and the Bergman space, have the same Fredholm symbol algebras and the same symbol homomorphism on generating operator, and the fact that the initial generators of these algebras are not unitary equivalent modulo compact operators.