Surveys in Mathematics and its Applications

ISSN1842-6298 (electronic), 1843-7265 (print)

Volume 3 (2008), 195 -- 209## A SURVEY ON DILATIONS OF PROJECTIVE ISOMETRIC REPRESENTATIONS

## Tania-Luminiţa Costache

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 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 [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].2000 Mathematics Subject Classification:20C25; 43A35; 43A65; 46L45; 47A20.

Keywords:multiplier; isometric projective representation; positive definite kernel; Kolmogorov decomposition; dilation.

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Tania-Luminiţa Costache

Faculty of Applied Sciences,

University "Politehnica" of Bucharest,

Splaiul Independenţei 313, Bucharest,

Romania.

e-mail: lumycos@yahoo.com