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In recent joint work with Felipe Flores, it was shown that the C*-algebra generated by the quasi-regular representation of Thompson's F < T is purely infinite. The proof involves an ultrapower technique developed by Robert and later considered by Ozawa. In this talk, we will discuss why ultrapowers are relevant in this context and how one works with them, focusing on a few proof techniques.

The local Langlands correspondence is a conjectural relation between the representations of a p-adic group and its L-parameters (which one can think of as representations of a `dual' group). Given a representation, it is natural to construct the hermitian dual representation. In this talk we present an explicit conjectural local Langlands correspondence, beyond the tempered case and including enhancements. We then show, assuming this conjecture, that constructing the hermitian dual corresponds to twisting its enhanced L-parameter by the compact real form on the dual group, generalising a result of Adams and Vogan. If time permits we will also show that standard desiderata of the local Langlands correspondence are in fact equivalent to this compatibility with hermitian duals.