Operator spaces and harmonic analysis

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Groups are amongst the most natural mathematical objects, developed to model symmetries. Infinite groups often have unruly algebraic behavior, which can frequently be tamed by examining their spatial behavior, i.e. using topology. Representation theory is is an important tool for understanding these topological groups. I am interested in studying particular mathematical objects, certain Banach algebras, which are associated with these representations. These algebras, coming from representation theory, are very closely related to operators (the natural framework in which representations are understood), and this connection allows for non-trivial spatial structures to be viewed in these algebras, which we hence call operator space structures. This is non-commutative harmonic analysis.