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Journal of Operator Theory

Volume 89, Issue 2, Spring 2023  pp. 587-601.

Minimal Stinespring representations of operator valued multilinear maps

Authors:  Erik Christensen
Author institution: Mathematics Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

Summary:  A completely positive linear map $\varphi$ from a $C^*$-algebra $\mathcal{A}$ into $B(\mathscr H)$ has a Stinespring representation as $\varphi(a) = X^*\pi(a)X,$ where $\pi$ is a $*$-representation of $\mathcal{A}$ on a Hilbert space $ \mathscr K$ and $X$ is a bounded operator from $\mathscr H $ to $\mathscr K. $ Completely bounded multilinear operators on $C^*$-algebras as well as some densely defined multilinear operators in Connes' noncommutative geometry also have Stinespring representations of the form $$ \Phi(a_1, \dots, a_k ) = X_0\pi_1(a_1)X_1 \cdots \pi_k(a_k)X_k$$ 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 boun-ded maps and for the geometrical maps, a natural minimality assumption implies that two such Stinespring representations have unitarily equivalent $*$-representations in their decompositions.

Keywords:  $C^*$-algebra, completely bounded, Stinespring representation, multilinear, nonncommutative geometry, unitarily equivalent

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