论文标题
Lebesgue的分解和Gleason-惠特尼（Whitney）财产的操作员代数
Lebesgue decompositions and the Gleason--Whitney property for operator algebras
论文作者
论文摘要
从广义上讲，本文与操作员代数的双空间有关。更确切地说，我们研究了我们所谓的Lebesgue预测的存在：操作员代数的双重预测，该预测检测到弱 - $*$ - $*$连续的双空间。与任何此类投影相关的是双重空间的Lebesgue分解。在操作员代数的综合背景下，我们对Lebesgue的预测特别感兴趣。我们展示了它们的存在如何与扩展特性密切相关，以使人联想到格里森和惠特尼的经典定理。 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.
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.
