Popova N. On configurations of subspaces in a Hilbert space

The thesis is devoted to the the study of the configurations of subspaces --- the collections of subspaces in a Hilbert space, where an angle between any two of subspaces is fixed. We obtain the conditions on the parameters-"angles", where configurations exist, that is, where the algebras have nontrivial *-representations. We study completely the configurations associated with a graph-cycle: the linear basis of algebra is described; the conditions on the arrangement of numbers on edges, where nontrivial *-representations exist, are obtained; the dimensions of irreducible *-representations are found and, in the pointed out bases of spaces of irreducible *-representations, we write out the matrices of operators corresponding to the generators. We find the conditions on one-parameter arrangement of numbers on the edges of cycle, where configurations exist. For tree the set of those values of an ''angle'' where the corresponding configurations exist is described by using the theory of graph spectra. New,more rich class of *-algebras is introduced. We prove the theorem about the equality of *-algebras from the new class to the *-algebras from previous class, where graph is a tree of some kind.