Marx, Gregory2017-07-182017-07-182017-07-17vt_gsexam:12415http://hdl.handle.net/10919/78353Positive 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.ETDIn CopyrightNoncommutative kernelsnoncommutative functionsuniformly bounded kernelscompletely positive kernelscompletely positive mapscompletely bounded mapsHilbert C*-modulesHilbert W*-modulesDissertation