Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation
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- by Igor Klep, Victor Vinnikov and Jurij Volčič PDF
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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 $\mathcal {O}^{\operatorname {ua}}_Y$ 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 $\mathcal {O}^{\operatorname {ua}}_Y$ with the factorization of $A$ over $\mathcal {O}^{\operatorname {ua}}_Y$. Different phenomena occur for a semisimple tuple of nonscalar matrices $Y$. Here it is shown that $\mathcal {O}^{\operatorname {ua}}_Y$ 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 $\mathcal {O}^{\operatorname {ua}}_Y$ does not embed into a skew field. Nevertheless, the ring $\mathcal {O}^{\operatorname {ua}}_Y$ 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.
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Additional Information
- Igor Klep
- Affiliation: Department of Mathematics, University of Ljubljana, Kongresni trg 12, 1000 Ljubljana, Slovenia
- Email: igor.klep@fmf.uni-lj.si
- Victor Vinnikov
- Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel
- MR Author ID: 224109
- Email: vinnikov@math.bgu.ac.il
- Jurij Volčič
- Affiliation: Department of Mathematics, Texas A&M University, 400 Bizzell Street, College Station, Texas 77843
- Email: volcic@math.tamu.edu
- Received by editor(s): December 4, 2019
- Published electronically: May 26, 2020
- Additional Notes: 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 - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5587-5625
- MSC (2010): Primary 32A20, 47A56, 16W60; Secondary 16R50, 32A05, 16K40
- DOI: https://doi.org/10.1090/tran/8076
- MathSciNet review: 4127886