Lebesgue decompositions and the Gleason--Whitney property for operator algebras

Broadly speaking, this paper is concerned with dual spaces of operator algebras. More precisely, we investigate the existence of what we call Lebesgue projections: central projections in the bidual of an operator algebra that detect the weak-$*$ continuous part of the dual space. Associated to any such projection is a Lebesgue decomposition of the dual space. We are particularly interested in Lebesgue projections in the context of inclusions of operator algebras. We show how their presence is intimately connected with an extension property for the inclusion reminiscent of a classical theorem of Gleason and Whitney. We illustrate that this Gleason--Whitney property fails in many examples of concrete operator algebras of functions, which partly explains why compatible Lebesgue decompositions are scarce, and highlights that the classical inclusion $H^\infty\subset L^\infty$ on the circle does not display generic behaviour.