Operator Theory Seminar
"Noncommutative Function Theory" is a subject that was first proposed by Joe Taylor in the late 1960's. It lay fallow until the turn of the century, when it exploded onto the scene thanks to contributions of Dan Voiculescu in free probability and Bill Helton in optimization theory. A full-blown presentation of it was developed by Dimitry Kaliuzhnyi-Verbovetskyi and Victor Vinnikov and published in their fundamental Foundations of Free Noncommutative Function Theory in 2014.
In 2013, Baruch Solel and I adapted the key ingredients of this subject to the setting of tensor algebras over $C^*$-algebras. In doing so, we realized that the underlying constructions in the subject were not presented in a functorial fashion. Efforts to reformulate them in a functorial way led us to the theory of $W^*$-categories. We have successfully reformulated the theory in terms of $W^*$-categories and this has led to a whole new set of possibilities to explore.
My plan for the two lectures is, first, to explain carefully the basic facts of noncommutative function theory as they are presented in the book mentioned above, and to show how it ineluctably leads one to the use of categories and functors. There is only one category used in that theory. However, when one tries to transfer that category to tensor algebras over $C^*$-algebras and the related function theory, one is led, again ineluctably, to reformulating everything in terms of $W^*$-categories. If time permits, I will show how our approach leads to a significant generalization of the so-called "noncommutative reproducing kernel Hilbert spaces" of Joe Ball, Greg Marx, and Victor Vinnikov.