A SURVEY ON DILATIONS OF PROJECTIVE ISOMETRIC REPRESENTATIONS

Abstract. In this paper we present Laca-Raeburn’s dilation theory of projective isometric representations of a semigroup to projective isometric representations of a group [4] and Murphy’s proof of a dilation theorem more general than that proved by Laca and Raeburn. Murphy applied the theory which involves positive definite kernels and their Kolmogorov decompositions to obtain the Laca-Raeburn dilation theorem [6]. We also present Heo’s dilation theorems for projective representations, which generalize Stine-spring dilation theorem for covariant completely positive maps and generalize to Hilbert C∗-modules the Naimark-Sz-Nagy characterization of positive definite functions on groups [2]. In the last part of the paper it is given the dilation theory obtained in [6] in the case of unitary operator-valued multipliers [3].