Minimal Stinespring Representations of Operator Valued Multilinear Maps

Abstract

A completely positive linear map $\phi$ from a C*-algebra $A$ into $B(H)$ has a Stinespring representation as $\phi (a)=X^{\ast }\pi (a)X,$ where $\pi$ is a *-representation of $A$ on a Hilbert space $K$ and $X$ is a bounded operator from $H$ to $K.$ Completely bounded multilinear operators on C*-algebras as well as some densely defined multilinear operators in Connes' non commutative geometry also have Stinespring representations of the form $$\Phi (a_1,\dots ,a_k)=X_0{\pi }_1(a_1)X_1\dots {\pi }_k(a_k)X_k$$ such that each $a_i$ is in a *-algebra $A_i$ and $X_0,\dots X_k$ are densely defined closed operators between the Hilbert spaces. We show that for both completely bounded maps and for the geometrical maps, a natural minimality assumption implies that two such Stinespring representations have unitarily equivalent *-representations in the decomposition.

17 pages