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 M-red. 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 star red on M-red from a strongly invariant star product star on M. The new questions which arc 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 star 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 star 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 (e infinity(M) [lambda], star)-submodule and a right e infinity (M-red)[lambda],star red)-submodule e(cf)infinity(C)[lambda] of C infinity(C)[lambda]; we define on it a e infinity(M-red) [lambda]-valued inneer product and e establish that this gives a strong Morita equivalence bimodule between e infinity(M-red)[lambda] and the finite rank operators on e(cf)infinity(C) [lambda] 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 (e infinity(M-red) [lambda],star red) on pre-Hilbert right D-modules to those of (e infinity(M)[lambda],star), for any auxiliary coefficient *-algebra D over C[lambda]. (C) 2010 Elsevier Inc All rights reserved.