The algebra of unbounded continuous functions on a Stonean space and unbounded operators
A space \(x\) is called extremely disconnected if the closure of every open set is also open, and is a Stonean space if it is also compact and Hausdorff. In this article the author studies the Kadison classes \(N(X)\), \(S(X)\) of normal and selfadjoint functions respectively, these being the unbounded continuous complex-valued and real-valued function on \(X\) respectively. The main results are: a simplified proof of Kadison's result that \(N(X)\) is an algebra, using a result on continuous extensions from open dense subsets of \(X\); an alternative proof of the result that \(S(X)\) is a boundedly complete lattice; a proof of the spectral theorem and characterization of the spectrum and the spectral projections of unbounded selfadjoint operators.
spectral projections of unbounded selfadjoint operators
scientific article; zbMATH DE number 1548303