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

References in corpus (1)

• Integration over compact quantum groups

Cited by in corpus (6)

• Noncommutative ball maps
• Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation
• Rational Cuntz states peak on the free disk algebra
• Free holomorphic functions on the unit ball of B(H)^n. II
• Matrix Extreme Points and Free extreme points of Free spectrahedra
• Free biholomorphic classification of noncommutative domains