  S. L. Woronowicz's theory of introducing C*-algebras generated by unbounded
elements is applied to q-normal operators satisfying the defining relation of
the quantum complex plane. The unique non-degenerate C*-algebra of bounded
operators generated by a q-normal operator is computed and an abstract
description is given by using crossed product algebras. If the spectrum of the
modulus of the q-normal operator is the positive half line, this C*-algebra
will be considered as the algebra of continuous functions on the quantum
complex plane vanishing at infinity, and its unitization will be viewed as the
algebra of continuous functions on a quantum 2-sphere.
