Absolutely dilatable bimodule maps
Alexandros Chatzinikolaou, Ivan G. Todorov, Lyudmila Turowska
TL;DR
This paper develops a noncommutative dilation theory for completely positive $\mathcal{D}'$-bimodule maps on $\mathcal{B}(H)$ by extending the Duquet–Le Merdy framework to include modularity over $\mathcal{D}'$ and a finite-tracial ancilla. It proves a full dilation characterization: a unital weak* continuous $\mathcal{D}'$-modular map is absolutely dilatable iff it admits a dilation implemented by a unitary $D\in \mathcal{D}\bar{\otimes}\mathcal{B}(K)$ returning to a finite-tracial $(\mathcal{N},\tau_{\mathcal{N}})$ with $\Phi(x)=(\mathrm{id}\otimes\tau_{\mathcal{N}})(D^*(x\otimes 1_{\mathcal{N}})D)$, equivalently described by the operator symbol $u_{\Phi}=(\mathrm{id}\otimes\mathrm{id}\otimes\tau_{\mathcal{N}})(D_{1,3}^*D_{2,3})$. Building on this, the authors introduce a hierarchy of dilatable maps—local, quantum, and approximately quantum—based on ancilla type and establish convexity and closure properties, with deep connections to the Connes Embedding Problem: whether the approximately quantum and quantum commuting classes coincide. In the commutative Schur multiplier setting, these notions align with measurable Schur multipliers, and the results illuminate how embeddability constraints govern factorisable maps. Overall, the work unifies noncommutative dilation theory with quantum-information concepts and ergodic theory, and explains how embeddability questions constrain the structure of dilatable, bimodule maps.
Abstract
We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.