Journal article
LOCAL THEORY OF FREE NONCOMMUTATIVE FUNCTIONS: GERMS, MEROMORPHIC FUNCTIONS, AND HERMITE INTERPOLATION
Transactions of the American Mathematical Society, v 373(8), pp 5587-5625
01 Aug 2020
Abstract
Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities.
This paper investigates the local theory of noncommutative functions. The first main result shows that for a scalar point Y, the ring O-Y(ua) of uniformly analytic noncommutative germs about Y is an integral domain and admits a universal skew field of fractions, whose elements are called meromorphic germs. A corollary is a local-global rank principle that connects ranks of matrix evaluations of a matrix A over O-Y(ua) with the factorization of A over O-Y(ua). Different phenomena occur for a semisimple tuple of nonscalar matrices Y. Here it is shown that O-Y(ua) contains copies of the matrix algebra generated by Y. In particular, there exist nonzero nilpotent uniformly analytic functions defined in a neighborhood of Y, and O-Y(ua) does not embed into a skew field. Nevertheless, the ring O-Y(ua) is described as the completion of a free algebra with respect to the vanishing ideal at Y. This is a consequence of the second main result, a free Hermite interpolation theorem: if f is a noncommutative function, then for any finite set of semisimple points and a natural number L there exists a noncommutative polynomial that agrees with f at the chosen points up to differentials of order L. All the obtained results also have analogs for (nonuniformly) analytic germs and formal germs.
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Details
- Title
- LOCAL THEORY OF FREE NONCOMMUTATIVE FUNCTIONS: GERMS, MEROMORPHIC FUNCTIONS, AND HERMITE INTERPOLATION
- Creators
- Igor Klep - University of LjubljanaVictor Vinnikov - Ben-Gurion University of the NegevJurij Volcic - Texas A&M University
- Publication Details
- Transactions of the American Mathematical Society, v 373(8), pp 5587-5625
- Publisher
- Amer Mathematical Soc
- Number of pages
- 39
- Grant note
- SCHW 1723/1-1 / Deutsche Forschungsgemeinschaft (DFG); German Research Foundation (DFG) J1-8132; N1-0057; P1-0222 / Slovenian Research Agency; Slovenian Research Agency - Slovenia Marsden Fund Council of the Royal Society of New Zealand; Royal Society of New Zealand; Marsden Fund (NZ)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000551418100010
- Scopus ID
- 2-s2.0-85090526124
- Other Identifier
- 991021861673904721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics