  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.
