TY - JOUR

T1 - LOCAL THEORY of FREE NONCOMMUTATIVE FUNCTIONS

T2 - GERMS, MEROMORPHIC FUNCTIONS, and HERMITE INTERPOLATION

AU - Klep, Igor

AU - Vinnikov, Victor

AU - Voľcǐc, Jurij

N1 - Funding Information:
Received by the editors December 4, 2019. 2010 Mathematics Subject Classification. Primary 32A20, 47A56, 16W60; Secondary 16R50, 32A05, 16K40. Key words and phrases. Free analysis, noncommutative function, analytic germ, universal skew field of fractions, noncommutative meromorphic function, Hermite interpolation. The first author was supported by the Slovenian Research Agency grants J1-8132, N1-0057, and P1-0222 and was partially supported by the Marsden Fund Council of the Royal Society of New Zealand. The second author was supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1. The third author was supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1.
Publisher Copyright:
© 2020 American Mathematical Society. All rights reserved.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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 Oua Y 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 Oua Y with the factorization of A over Oua Y . Different phenomena occur for a semisimple tuple of nonscalar matrices Y. Here it is shown that Oua Y 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 Oua Y does not embed into a skew field. Nevertheless, the ring Oua Y 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.

AB - 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 Oua Y 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 Oua Y with the factorization of A over Oua Y . Different phenomena occur for a semisimple tuple of nonscalar matrices Y. Here it is shown that Oua Y 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 Oua Y does not embed into a skew field. Nevertheless, the ring Oua Y 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.

KW - Analytic germ

KW - Free analysis

KW - Hermite interpolation

KW - Noncommutative function

KW - Noncommutative meromorphic function

KW - Universal skew field of fractions

UR - http://www.scopus.com/inward/record.url?scp=85090526124&partnerID=8YFLogxK

U2 - 10.1090/tran/8076

DO - 10.1090/tran/8076

M3 - Article

AN - SCOPUS:85090526124

SN - 0002-9947

VL - 373

SP - 5587

EP - 5625

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 8

ER -