Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden–Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product ⋆red on Mred from a strongly invariant star product ⋆ on M. The new questions which are addressed in this paper concern the existence of natural ∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ∗-algebra.We assume that ⋆ is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J−1(0), with some equivariance property, defines a ∗-involution for ⋆red on the reduced space. Looking into the question whether the corresponding ∗-involution is the complex conjugation (which is a ∗-involution in the Marsden–Weinstein context) yields a new notion of quantized modular class.We introduce a left (C∞(M)〚λ〛,⋆)-submodule and a right (C∞(Mred)〚λ〛,⋆red)-submodule of C∞(C)〚λ〛; we define on it a C∞(Mred)〚λ〛-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C∞(Mred)〚λ〛 and the finite rank operators on . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ∗-representations of (C∞(Mred)〚λ〛,⋆red) on pre-Hilbert right D-modules to those of (C∞(M)〚λ〛,⋆), for any auxiliary coefficient ∗-algebra D over C〚λ〛.