Thick Elements and States in \(C^*\)-Algebras in View of Frame Theory

Abstract

We study some classes of noncommutative \(C^*\)-algebras and generalize some results which were originally obtained for commutative algebras in topological terms. In particular, we are interested in results obtained for topological spaces with properties close to separability and \( \sigma \)-compactness. To obtain the algebraic, noncommutative versions of corresponding properties, we define and use the notions of thick elements and states. In particular, an element is thick if the only element orthogonal to it is zero.