An algebra of functions on q-deformed Anti-de Sitter space AdS(q)(D) With s
tar-structure is defined for roots of unity, which is covariant under U-q(s
o(2, D - 1)). The scalar fields have an intrinsic high-energy cutoff, and a
rise most naturally on products of the quantum AdS space with a classical s
phere. Hilbert spaces of scalar fields are constructed.