2000/05/23 by Schmuedgen, Konrad · 1 citation
#16W30 (Primary) 46L87 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.math/0005225
Suppose q is a complex number of modulus one and different from 1,-1. Let O(R2q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R2q) on Hilbert space and the Weyl calculus of pseudodifferential operators we construct *-algebras of ``functions'' on the quantum quarter plane R++q and on the real quantum plane R2q which are left module *-algebras for the Hopf *-algebra Uq(gl2(R)). We define a family hk, k∈ Z2, of covariant positive linear functionals on these *-algebras and study the actions of the *-algebras O(R2q) and Uq(gl2(R)) 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. A shorter version appeared in International Journal of Mathematics 13 (2002), 279-321.