Minimal Stinespring Representations of Operator Valued Multilinear Maps

A completely positive linear map $\varphi$ from a C*-algebra $A$ into $B(H)$ has a Stinespring representation as $\varphi(a) = X^*\pi(a)X,$ 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.