Non-commutative boundaries and peaking phenomena for operator algebras

A classical problem in complex function theory is to identify the unique smallest boundary associated with a subalgebra of continuous functions on a compact Hausdorff space — the Choquet boundary, which the works of Bishop, Choquet, and Glicksberg interpret via unique representing measures, peak points, and peak projections. We investigate their work in a non-commutative framework and introduce a relationship between these characterizations for unital subalgebras of bounded linear operators. This yields a notion of non-commutative boundary and a characterization of minimality for the non-commutative Choquet boundary — which is not always minimal, though natural examples from multivariate operator theory show it can be in special cases. Joint work with Raphaël Clouâtre.