Non-commutative measure theory: Henkin and analytic functionals on $$\mathrm {C}^*$$-algebras

Abstract

Henkin functionals on non-commutative $$\mathrm {C}^*$$ -algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg–König–Seever decomposition of the dual space of a $$\mathrm {C}^*$$ -algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of states. Leveraging this tool, we show that Henkin functionals are absolutely continuous with respect to the so-called analytic functionals if and only if a certain compatibility condition is satisfied by the ambient weak- $$*$$ topology. In contrast with the classical setting, the issue of stability under absolute continuity is not automatic in this non-commutative framework, and we illustrate its key role in sharpening our description of Henkin functionals. Our machinery yields new insight when specialized to the multiplier algebras of the Drury–Arveson space and of the Dirichlet space, and to Popescu’s noncommutative disc algebra. As another application, we make a contribution to the theory of non-commutative peak and interpolation sets.