# Dilations for $C^ast$-dynamical systems with abelian groups on Hilbert $C^ast$-modules

Document Type: Research Paper

Authors

College of Mathematics and Information Science‎, ‎Shaanxi Normal University‎, ‎Xi'an‎, ‎Shaanxi‎, ‎710062‎, ‎P‎.‎R‎. ‎China.

Abstract

‎In this paper we investigate the dilations of completely positive definite representations‎ ‎of $C^\ast$-dynamical systems with abelian groups on Hilbert $C^\ast$-modules‎. ‎We show that if $(\mathcal{A}‎, ‎G,\alpha)$ is a $C^\ast$-dynamical system with $G$ an abelian group‎, ‎then every completely positive definite covariant representation $(\pi,\varphi,E)$ of $(\mathcal{A}‎, ‎G,\alpha)$ on a Hilbert $C^\ast$-module $E$ admits an unitary dilation $((\hat{\pi},\hat{\varphi},\hat{E})).$

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### History

• Receive Date: 10 June 2015
• Revise Date: 21 March 2016
• Accept Date: 29 May 2016