Projective Unitary Representations And Their Frame Duality Properties

This thesis mainly studies the group projective unitary representations and their frame duality properties.The thesis consists of three chapters.In chapter one,we introduce the yon Neumann algebras of group projective unitary representations.Relying on group representations and group-like unitary system,as well as using the convolution which we reconstruct,we present a new characteristic of a group projective unitary representation.Making use of this characteristic,we prove the finiteness of the von Neumann algebra of group projective unitary representation.In addition,we give the condition which make it to be a factor and the relation of left and right yon Ncumann algebras.Furthermore, we depict the specific form of central elements of the von Neumann algebra.This method is different from Han Deguang's that is used in group-like unitary system.In chapter two,we study the frame duality properties of projective unitary representations.We prove two weak equivalences of projections are consistent. One weak equivalence of two projections is in the von Neumann algebra of projectivc unitary rcprcscntation with multiplicrμ,the other weak equivalence of two corresponding projections is in the von Neumann algebra of left regularμ-projective unitary representation.This result generalizes the frame duality properties of projective unitary representation written by Han Deguang.Here we use the method that from special to general.In chapter three,we mainly introduce best approximation of abstract wavelet system and give the detailed proof of Han Deguang's best approximation theorem. For complete frame vectorψof group-like unitary system,its best approximation is S^-1/2ψ,where S is frame operator.Although this is not true for abstract wavelet system,we give the similar result of semi-orthogonal complete frame vector and its complete proof.