Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation

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 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 O ua Y with the factorization of A over O ua Y . Different phenomena occur for a semisimple tuple of non-scalar matrices Y . Here it is shown that there exist nonzero nilpotent uniformly analytic functions defined in a neighborhood of Y . In particular, O ua Y does not embed into a skew field. Nevertheless, the ring O ua Y is described as the completion of a free algebra with respect to the vanishing ideal at Y . This is the 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 (non-uniformly) analytic germs and formal germs.

Blaschke-Singular-Outer factorization of free non-commutative functions

“…In other words, if H is not identically zero, then one cannot have detH(Z) = 0 for all Z ∈ B d N . Indeed, by [29,Theorem 5.7] the inner rank of H considered as a 1 × 1 matrix over the ring of germs of uniformly analytic NC functions at 0 is given by max n rank(H(Z)) n Z ∈ a neighbourhood of 0 ∩ B d n . This latter number is less than 1 since det H(Z) = 0 for every Z ∈ B d N .…”

Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity

“…Agler, McCarthy and Young, in the process of investigating noncommutative symmetric functions in two variables, developed an intricate theory of noncommutative manifolds [5,3]. On the other hand, Klep, Vinnikov and Volcic took an approach from germ theory [32]. The Free Universal Monodromy theorem says, in the case of free sets, that the theory arising from analytic continuation is essentially trivial.…”

“…Free Universal Monodromy implies various corollaries, such as the free inverse function theorem [28,38,4,33,34] and the universal existence of pluriharmonic conjugates. Furthermore, it shows any cohomology theory arising from sheaf theory of free analytic functions, as has been developed from differing angles [5,3,32], may be trivial on free sets.…”

Free noncommutative principal divisors and commutativity of the tracial fundamental group

“…We define the tracial covering space to be the set of paths in a domain originating at some fixed base point which can be distinguished via analytic continuation, and show that the corresponding group generated by loops, the tracial fundamental group is abelian. Prior developments in sheaf theory were made for free noncommutative functions in Klep, Volcic, Vinnikov [15], in the study of noncommutative symmetric functions by Agler and Young [1], the noncommutative manifold theory Agler, McCarthy and Young [3], implicit function theorems of Agler and McCarthy [2], and free universal monodromy [17].…”