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 \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. 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 $\m^2$ we construct a 2-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.

75 pages, 1 figure. Appeared in Trans. London Math. Soc.5, no. 1, [2018] 132:183. Updated post to add an extra hypothesis in the statement of Prop. 3.4. (This error appears in the published version.)

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• Dilations, inclusions of matrix convex sets, and completely positive maps
• Free convex sets defined by rational expressions have LMI representations
• The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
• Matrix Convex Hulls of Free Semialgebraic Sets
• Global holomorphic functions in several non-commuting variables II
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Cited by in corpus (4)

• Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity
• Noncommutative rational functions invariant under the action of a finite solvable group
• Free noncommutative principal divisors and commutativity of the tracial fundamental group
• Weak-* and completely isometric structure of noncommutative function algebras