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Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity
arXiv (Cornell University)
18 Feb 2020
Abstract
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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Details
- Title
- Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity
- Creators
- J. E Pascoe
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021880192704721