Entanglement in C$^*$-algebras: tensor products of state spaces
Abstract
We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets arising as state spaces of C-algebras, and, relatedly, study entanglement in (infinite dimensional) C-algebras. The minimal Namioka-Phelps tensor product of the state spaces of two C-algebras is shown to correspond to the set of separable (= un-entangled) states on the tensor product of the C-algebras. We show that these maximal and minimal tensor product of the state spaces agree precisely when one of the two C-algebras is commutative. This confirms an old conjecture by Barker in the case where the compact convex sets are state spaces of C-algebras. The Namioka-Phelps tensor product of the trace simplexes of two C-algebras is shown always to be the trace simplex of the tensor product of the C-algebras. This can be used, for example, to show that the trace simplex of (any) tensor product of C-algebras is the Poulsen simplex if and only if the trace simplex of each of the C-algebras is the Poulsen simplex or trivial (and not all trivial).
Cite
@article{arxiv.2512.10410,
title = {Entanglement in C$^*$-algebras: tensor products of state spaces},
author = {Magdalena Musat and Mikael Rørdam},
journal= {arXiv preprint arXiv:2512.10410},
year = {2026}
}
Comments
26 pages. The statements of Corollaries 4.2 and 4.4 have been corrected. New reference added. Incorrect statement in the proof of Theorem 3.15 has been corrected