Involutions and representations for reduced quantum algebras

Abstract

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 ⋆ red on M red 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 ∞ ( M red ) 〚 λ 〛 , ⋆ red ) -submodule C cf ∞ ( C ) 〚 λ 〛 of C ∞ ( C ) 〚 λ 〛 ; we define on it a C ∞ ( M red ) 〚 λ 〛 -valued inner product and we establish that this gives a strong Morita equivalence bimodule between C ∞ ( M red ) 〚 λ 〛 and the finite rank operators on C cf ∞ ( C ) 〚 λ 〛 . 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 ∞ ( M red ) 〚 λ 〛 , ⋆ red ) on pre-Hilbert right D -modules to those of ( C ∞ ( M ) 〚 λ 〛 , ⋆ ) , for any auxiliary coefficient ∗-algebra D over C 〚 λ 〛 .