Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

Abstract

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d‐dimensional nc‐universe Md. We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables.

A refined nc Oka–Weil theorem

“…In this respect, Pascoe already pointed out the same kind of flaw in the free topology in [8]. Moreover, Agler et al pointed out another flaw, that is, the free topology is not Hausdorff [2, Corollary 7.6 and Proposition 7.13].…”

“…We will briefly explain a cryptic phenomenon of free holomorphic functions from this viewpoint. Pascoe [8] proved that if , then there is an entire free holomorphic function (hence, it is free continuous due to [2, Proposition 3.8]) which is unbounded on the row ball. This phenomenon never occurs in the classical setting.…”

Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity

“…In free noncommutative function theory, there have been at least two recent approaches to the development of sheaf theory. 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].…”

“…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].…”