Free Analysis Questions II: The Grassmannian Completion and The Series Expansions at the Origin

Abstract

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.

88 pages; Subsections 12.2, 12.3, 12.4, 12.5 and 13.9 have been added. The rest are only minor changes and corrections. Added subsection 13.10 and made additions to 3.9, Appendix II and to the References