A survey on dilations of projective isometric representations

In this paper we present Laca-Raeburn's dilation theory of projective isometric representations of a semigroup to projective isometric representations of a group [M.Laca and I.Raeburn, Proc. Amer. Math. Soc., 1995] 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 [G.J. Murphy, Proc. Amer. Math.Soc., 1997]. We also present Heo's dilation theorems for projective representations, which generalize Stinespring dilation theorem for covariant completely positive maps and generalize to Hilbert C*-modules the Naimark-Sz-Nagy characterization of positive definite functions on groups [J.Heo, J.Math.Anal.Appl., 2007]. In the last part of the paper it is given the dilation theory obtained in [G.J. Murphy, Proc. Amer. Math.Soc., 1997] in the case of unitary operator-valued multipliers [Un Cig Ji, Young Yi Kim and Su Hyung Park, J. Math. Anal. Appl., 2007].