Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 9 (2014), 149 -- 166

A COVARIANT STINESPRING TYPE THEOREM FOR τ-MAPS

Harsh Trivedi

Abstract. Let τ be a linear map from a unital C*-algebra A to a von Neumann algebra B and let C be a unital C*-algebra. A map T from a Hilbert A-module E to a von Neumann C-B module F is called a τ-map if

< T(x), T(y) >=τ( < x, y > ) for all x,y∈ E.
A Stinespring type theorem for τ-maps and its covariant version are obtained when τ is completely positive. We show that there is a bijective correspondence between the set of all τ-maps from E to F which are (u',u)-covariant with respect to the dynamical system (G,η,E) and the set of all (u' ,u)-covariant τ~-maps from the crossed product E×η G to F, where τ and τ~ are completely positive.

2010 Mathematics Subject Classification: Primary: 46L08, 46L55; Secondary: 46L07, 46L53.
Keywords: Stinespring representation; Completely positive maps; Von Neumann modules; Dynamical systems.

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Harsh Trivedi
Department of Mathematics, Indian Institute of Technology Bombay,
Powai, Mumbai-400076,
India.
E-mail: harsh@math.iitb.ac.in

http://www.utgjiu.ro/math/sma