Topological group actions and associated Banach algebras

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My research is focused on the interplay between topological groups and certain classes of Banach algebras, typically algebras whose structures are intrinsically related to the properties and dynamics of the underlying group. All of the research in which I can see myself active builds upon existing research accomplishments.  Much of the insight will be gained through the study of the structure of Fourier-Stieltjes, measure algebras, certain fixed point subalgebras, and to similarity problems around group and Fourier algebras.  At many stages, there is necessity to move beyond locally compact groups. Fourier-Stieltjes (FS) algebras My new angle for studying FS algebras are the Eberlein-de Leeuw-Glicksberg (EdLG) decompositions of my recent preprint [4]. These decompositions arise from information of the dynamics of how groups act on Hilbert spaces. They have given a new inroad in examining operator amenability of FS algebras. Furthermore, they open the door to studying certain classes of non-locally compact groups, which often arise naturally, even when the underlying group is locally compact. Measure algebras Measure algebras beautifully encode much algebraic and topological information about locally compact groups, but are magnificent and daunting in their complexity. My view is that they are dual objects to FS algebras in a manner generalizing Pontryagin duality. Understanding the spine, or the analogue of EdLG decompositions will open a door to learn more deeply about their structures. In particular, insight may be gained into structures of idempotent/projections and of invertibles, and also into certain classes of non-locally compact groups. Fixed point subalgebras The study of the algebra of fixed points of a subalgebra of an FS or measure algebra can simplify the understanding of the algebra, but also show underlying complexities. It also invites new classes of tools into the study, e.g. hypergroups. Some difficult open problems about amenability remain which I either wish to directly attack, or seek hints by looking at analogous problems. Similarity problems Given a group (i.e. its L^1-algebra) or a Fourier algebra, we wish to understand under what circumstances any suitably bounded representation factors through a C*-algebra. I wish to continue determining classes of locally compact groups for which Fourier algebras enjoy this property. The classes of problems give many entry points, of varying difficulty, for training of students and post-docs.  Canada maintains a rich tradition of research in non-commutative harmonic analysis, which links to related fields such as operator algebras, in which we are strong.  Training in these fields has produced many good researchers, and has helped seed, through talent and research accomplishments, our leading position in such fields as quantum information theory, which is well-established at my university.