Abstract
Suppose q≠±1 is a complex number of modulus one. Let [Formula: see text] be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using Hilbert space representations of [Formula: see text] and the Weyl calculus of pseudodifferential operators we construct *-algebras of "functions" on the quantum quarter plane [Formula: see text] and on the real quantum plane [Formula: see text] which are left module *-algebras for the Hopf *-algebra [Formula: see text]. We define covariant positive linear functionals hk, k∈ℤ2, and study the actions of the *-algebras [Formula: see text] and [Formula: see text] on the associated Hilbert spaces. Quantum analogs of the partial Fourier transforms and the Fourier transform are found. A differential calculus on the "function" *-algebras is also developed and investigated.
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