Analytic functions relative to a covariance map $\eta$: I. Generalized Haagerup products and analytic relations

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.