Minimal Stinespring Representations of Operator Valued Multilinear Maps

2021/08/26 by Christensen, Erik
#46L07 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2108.11778

Abstract

A completely positive linear map φ from a C*-algebra A into B(H) has a Stinespring representation as φ(a) = X^*π(a)X, where π 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 Φ(a₁, …, a_k ) = X₀π₁(a₁)X₁ … π_k(a_k)X_k such that each a_i is in a *-algebra A_i and X₀, … 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.

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