Function algebras and operator algebras arising in noncommutative harmonic analysis
Classical Fourier analysis on the circle, or for periodic functions on the line, rests on the duality between spatial symmetries (rotations) and symmetries on the frequency side (shifts). In more general situations the spatial symmetries need not commute with each other: think of the rotation group in three or more dimensions, or the group generated by translations and dilations on the line. Generalizing Fourier analysis to this setting leads to two kinds of algebraic structures: operator algebras generated by group representations (living on the frequency side); and function algebras generated by the "matrix coefficients" of these representations (living on the spatial side). The bridge between the two sides is a suitable generalization of the classical Fourier transform and Plancherel theorem. For the two examples mentioned above, this abstract theory underlies tools such as spherical harmonics and the wavelet transform.
A PhD project in this area would investigate structural properties of these algebras using methods from functional analysis and representation theory, with a focus on understanding examples attached to particular families of groups. Within this broad remit there are distinct but related possible topics of research. One possible direction is to study the local structure of L^{p}-operator algebras generated by representations of Lie groups, where for p ≠ 2 much less is known than for C^{*}-algebras. A different direction would attempt to compute or estimate cohomological invariants associated to function algebras on Lie groups; here is there considerable scope for computer-aided experimentation, although rigorous proofs will ultimately be required.
For some of my own work in these directions, see the references below.
M. Eugenia Celorrio started a PhD at Lancaster in 2018 with H. Garth Dales as principal supervisor, in the general area of Banach function algebras and Arens (ir)regularity. I have a co-supervisory role.
Blake Green started a PhD with me in 2017, studying some particular examples of subhomogeneous Banach and operator algebras. Among other things, Blake has considered certain 2-subhomogeneous examples arising from almost disjoint families in $\mathcal P(\mathbb N)$, and explicitly calculated the Fell topology on the space of irreducible representations for such an algebra. Some partial results have also been obtained concerning amenable closed subalgebras of $C(X; {\mathbb M}_2)$ for certain countable compact $X$. Currently we are looking at some cb-multiplier questions arising when considering certain ideals in the function algebra $C^1[0,1]$.
Bence Horváth obtained his PhD in June 2019. His thesis is titled Algebras of operators on Banach spaces and homomorphisms thereof and was co-supervised by me and Niels Laustsen. Some of the work from his thesis can be found on the arXiv as 1807.10578 and 1811.06865 (both published recently). The second of these papers introduces and studies what Bence calls the "SHAI property for a Banach space E", which is the condition that every surjective algebra homomorphism ${\mathcal B}(E)\to {\mathcal B}(F)$ must be injective, provided $F$ is not the zero space. Bence established this property for several classical and not so classical spaces, and has added to the list in subsequent work with Kania (arXiv 2007.14112); it has been also verified to hold for $L_p(\mu)$-spaces by Johnson, Phillips and Schechtman (arXiv 2102.03966).
Chris Menez obtained his PhD in August 2019, having successfully defended his thesis in November 2018. His thesis is titled A category-theoretic approach to extensions of Banach algebras, and was based on the problem of extending classical ideas of Busby to the more general situation where the ideal in the extension is faithful (hence not covered by the glut of results on singular extensions) but is not closed in its multiplier algebra (hence not covered by Busby's work, nor by the more recent work of Blecher & Royce).
Previously, at the University of Saskatchewan I supervised two graduate students to completion: one PhD student and one Master's student, both co-supervised with Ebrahim Samei: