I was a researcher in functional analysis, specifically operator spaces, operator algebras and C*-algebraic quantum groups. I also taught a wide variety of courses from calculus through to graduate-level functional analysis.
These are notes that I wrote for courses that I taught when I was in academia. They are provided in case anyone finds them useful. The notes are fairly complete, but don’t have bibliographic references - frequently the primary source for proofs were my notes from lectures I attended as a student.
Real Analysis is a set of notes that I used when I taught graduate-level real analysis. This is a full-year set of notes which covers measure theory, general topology and introductory functional analysis.
Modern Group Theory is a set of notes that I used when I taught undergraduate abstract agebra. This is a single semester set of notes which is an introduction to group theory. The LaTeX source for these notes is available on github.
The following papers and posters by myself and my co-authors are available on this site:
S.J. Kern, J. Portman, C. Webster, I. Cernyte, Y. Kiridooshi, W. Suda, C. Mueller, R. Cardwell, Natural Language Processing Tool for Automated Curation and Quality Assessment of Reference Databases; A Case Study Using 16S rRNA Respositories, Presented at the 69th Annual Meeting of the American Society of Human Genetics, October 16, 2019, Houston, Texas. (PDF)
We provide a geometric condition which says precisely when the metric boundary of a proper separable metric space has no Busemann points. This result generalizes an earlier result which applied only to graph metrics.
We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient of the metric boundary and the quotient map is continuous, and that therefore a word-hyperbolic group has an amenable action on the metric boundary of its Cayley graph. Furthermore, if the space is 0-hyperbolic, the boundaries agree, and as a consequence there are no non-Busemann points on the boundary of such spaces. These results have significance for the study of Lip-norms on group C*-algebras.
We provide a geometric condition which determines whether or not every point on the metric boundary of a graph with the standard path metric is a Busemann point, that is it is the limit point of a geodesic ray. We apply this and a related condition to investigate the structure of the metric boundary of Cayley graphs. We show that groups such as the braid group and the discrete Heisenberg group have boundary points of the Cayley graph which are not Busemann points when equipped with their usual generators.
We show that the multipliers of Pedersen’s ideal of a C*-algebra A correspond to the densely defined operators on A which are affiliated with A in the sense defined by Woronowicz, and whose domains contain Pedersen’s ideal. We also extend the theory of q-continuity developed by Akemann to unbounded operators and show that these operators correspond to self-adjoint operators affiliated with A.
We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C-convex sets of complex matrices and the matrix state spaces of C-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set K may be thought of as the state space of the space of continuous affine functions on K.
Edward G. Effros and Corran Webster, Operator Analogues of Locally Convex Spaces, Operator Algebras and Applications (Samos 1996), (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 495), Kluwer, 1997 (DVI) (PDF) (PostScript)
Local operator spaces are defined to be projective limits of operator spaces. These limits arise when one considers linear spaces of unbounded operators, and may be regarded as the "quantized" or "operator" analogues of locally convex spaces. It is shown that for nuclear spaces, the maximal and minimal quantizations coincide. Thus, in a striking contrast to normed spaces, a nuclear space has precisely one quantization. Furthermore, it is shown that a local operator space is nuclear in the operator sense if and only if its underlying locally convex space is nuclear. Operator versions of bornology and duality are also considered.