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Operator noncommutative functions

Part of: Selfadjoint operator algebras General theory of linear operators

Published online by Cambridge University Press:  24 May 2022

Meric Augat Open the ORCID record for Meric Augat [Opens in a new window]  and
John E. McCarthy Open the ORCID record for John E. McCarthy [Opens in a new window]
Meric Augat*
Affiliation:
Washingston University in St. Louis, St. Louis, MO, USA e-mail: mccarthy@wustl.edu
John E. McCarthy
Affiliation:
Washingston University in St. Louis, St. Louis, MO, USA e-mail: mccarthy@wustl.edu
*
e-mail: maugat@wustl.edu
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Abstract

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.

Keywords

NC functionsfree analysisfree polynomialsoperator-valued functions

MSC classification

Primary: 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 492 - 508
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This research was partially supported by the National Science Foundation Grant DMS 2054199.

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