Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 3 (2008), 195 -- 209
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  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 .
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 .
In the last part of the paper it is given the dilation theory obtained in  in the case of unitary operator-valued multipliers .2000 Mathematics Subject Classification: 20C25; 43A35; 43A65; 46L45; 47A20.
Keywords: multiplier; isometric projective representation; positive definite kernel; Kolmogorov decomposition; dilation.
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Faculty of Applied Sciences,
University "Politehnica" of Bucharest,
Splaiul Independenţei 313, Bucharest,