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

Abstract

Henkin functionals on non-commutative ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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-$\ast$ 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.

33 pages