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Virginia Tech

1. Marx, Gregory. Noncommutative Kernels.

Degree: PhD, Mathematics, 2017, Virginia Tech

Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels. Advisors/Committee Members: Ball, Joseph A (committeechair), Rossi, John F (committee member), Floyd, William J (committee member), Elgart, Alexander (committee member).

Subjects/Keywords: Noncommutative kernels; noncommutative functions; uniformly bounded kernels; completely positive kernels; completely positive maps; completely bounded maps; Hilbert C*-modules; Hilbert W*-modules

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APA (6th Edition):

Marx, G. (2017). Noncommutative Kernels. (Doctoral Dissertation). Virginia Tech. Retrieved from

Chicago Manual of Style (16th Edition):

Marx, Gregory. “Noncommutative Kernels.” 2017. Doctoral Dissertation, Virginia Tech. Accessed July 20, 2018.

MLA Handbook (7th Edition):

Marx, Gregory. “Noncommutative Kernels.” 2017. Web. 20 Jul 2018.


Marx G. Noncommutative Kernels. [Internet] [Doctoral dissertation]. Virginia Tech; 2017. [cited 2018 Jul 20]. Available from:

Council of Science Editors:

Marx G. Noncommutative Kernels. [Doctoral Dissertation]. Virginia Tech; 2017. Available from: