  The fully matricial generalization in part I, of the difference quotient
derivation on holomorphic functions, in which ${\mathbb C}$ is replaced by a
Banach algebra $B$, is extended from the affine case to a Grassmannian
completion. The infinitesimal bialgebra duality, the duality transform
generalizing the Stieltjes transform and the spectral theory with non-commuting
scalars all extend to this completion. The series expansions of fully matricial
analytic functions are characterized, providing a new way to generate fully
matricial functions.
