  We generalize module weak-* Haagerup tensor products to obtain complete
quotients of normal Haagerup tensor product included in canonical Hilbert
spaces associated to completely positive normal (covariance) maps $\eta$ on a
finite von Neumann algebra $B$. We construct in this way dual operator spaces,
providing new examples even in the case of module extended Haagerup tensor
products. This is the basis for defining a matrix normed algebra of analytic
functions that captures the relations of free semicircular variables with
covariance $\eta$. We prove that a class of non-commutative random variables
having finite Fisher information relative to $\eta$ have also no analytic
relations among our class of analytic functions.
