On representations of completely bounded maps (Q1173811)

Let \(A\) be a \(C^*\)-algebra, \(H\) a Hilbert space, and \(L: A\to B(H)\) a completely bounded map. The author shows that \(L\) can be represented in the form \(L(a)=V^*T\pi(a)V\), where \(V: H\to M\) is an isometry to a certain Hilbert space \(M\), \(T\in B(M)\), and \(\pi: A\to B(M)\) is a unital \(^*\)-representation. [This is derived from (and equivalent to) the representation obtained by \textit{V. Paulsen}, Proc. Amer. Math. Soc. 86, 91-96 (1982; Zbl 0554.46028).] Further, the author studies unitary equivalence of two such representations: If \((\pi,V,T,M)\) and \((\pi',V',T',M')\) are minimal, then there exists a unitary operator \(U: M\to M'\) such that \(UV=V'\), \(U\pi U^*=\pi'\) and \(UTU^*=T'\), if and only if \(V^*\pi V=V'{}^*\pi'V'\).