Unitary multiplier and dilation of projective isometric representation

In this paper we study unitary operator-valued multiplier σ on a normal subsemigroup S of a group G with its extension to G. A dilation of a projective isometric σ-representations of S to a projective unitary Φ(σ)-representation of G is established for a suitable unitary operator-valued multiplier Φ(σ) associated with the multiplier σ which is explicitly constructed during the study.