An analog of a modular functor from quantized Teichmüller theory, I

(i) b is a deformation parameter, related to the traditional ~ via b = √ ~, (ii) Σ is a two-dimensional topological surface with finitely many boundary components, (iii) HT b (Σ) is a Hilbert-space (possibly infinite-dimensional), (iv) OTb (Σ) is an algebra of bounded operators on HT b (Σ) and, (v) MTb (Σ) is a unitary projective representation of the mapping class group of Σ on HT b (Σ). The data ( OTb (Σ),M T b (Σ) ) are restricted by the requirement that a suitably defined limit of OTb (Σ) for b → 0 should reproduce the commutative algebra of functions on the Teichmüller space T (Σ), whereas a natural limit b → 0 of the automorphisms of OTb (Σ) which are induced by the representation MTb (Σ) should correspond to the natural action of the mapping class group MC(Σ) on T (Σ).