Multipliers of Hilbert algebras and deformation quantization

Abstract

In this paper, we introduce the notion of multiplier of a Hilbert algebra. The space of bounded multipliers is a semifinite von Neumann algebra isomorphic to the left von Neumann algebra of the Hilbert algebra, as expected. However, in the unbounded setting, the space of multipliers has the structure of a *-algebra with nice properties concerning commutant and affiliation: it is a pre-GW*-algebra. And this correspondence between Hilbert algebras and its multipliers is functorial. Then, we can endow the Hilbert algebra with a nice topology constructed from unbounded multipliers. As we can see from the theory developed here, multipliers should be an important tool for the study of unbounded operator algebras. We also formalize the remark that examples of non-formal deformation quantizations give rise to Hilbert algebras, by defining the concept of Hilbert deformation quantization (HDQ) and studying these deformations as well as their bounded and unbounded multipliers in a general way. Then, we reformulate the notion of covariance of a star-product in this framework of HDQ and multipliers, and we call it a symmetry of the HDQ. By using the multiplier topology of a symmetry, we are able to produce various functional spaces attached to the deformation quantization, like the generalization of Schwartz space, Sobolev spaces, Gracia-Bondia-Varilly spaces. Moreover, the non-formal star-exponential of the symmetry can be defined in full generality and has nice relations with these functional spaces. We apply this formalism to the Moyal-Weyl deformation quantization and to the deformation quantization of Kahlerian Lie groups with negative curvature.