Inverse and implicit function theorems for noncommutative functions on operator domains

Mancuso, Mark E.

doi:10.7900/jot.2018oct21.2237

Cited by 3 publications

(1 citation statement)

References 14 publications

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“…Such functions were studied in [2, 19]. A key assumption in those papers, however, was that the function was also sequentially continuous in the strong operator topology.…”

Section: Introductionmentioning

confidence: 99%

Operator noncommutative functions

Augat¹,

McCarthy²

2022

Can. Math. Bull.

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We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

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“…Such functions were studied in [2, 19]. A key assumption in those papers, however, was that the function was also sequentially continuous in the strong operator topology.…”

Section: Introductionmentioning

confidence: 99%

Operator noncommutative functions

Augat¹,

McCarthy²

2022

Can. Math. Bull.

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Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Pascoe

2023

Tohoku Math. J. (2)

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Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity

Pascoe¹

2020

Preprint

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We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets. We describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and solve a related problem of Martin-Shamovich. We generalize the Dym-Helton-Klep-McCullough-Volcic theorem-a uniformly real analytic free noncommutative function is plurisubharmonic if and only if it can be written as a composition of a convex function with an analytic function. The decomposition is essentially unique. The result is first established locally, and then Free Universal Monodromy implies the global result. Moreover, we see that plurisubharmonicity is a geometric property-a real analytic free function plurisubharmonic on a neighborhood is plurisubharmonic on the whole domain. We give an analytic Greene-Liouville theorem, an entire free plurisubharmonic function is a sum of hereditary and antihereditary squares.

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Inverse and implicit function theorems for noncommutative functions on operator domains

Cited by 3 publications

References 14 publications

Operator noncommutative functions

Operator noncommutative functions

Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity

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