Thick elements and states in $C^*$-algebras in view of frame theory

We study some classes of noncommutative $C^*$-algebras and generalize some results on the absence of frames (standard and non-standard) 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.