Entanglement in C$^*$-algebras: tensor products of state spaces

• URL: http://arxiv.org/abs/2512.10410v2
• Date: Sat, 13 Dec 2025 17:43:07 GMT
• Title: Entanglement in C$^*$-algebras: tensor products of state spaces
• Abstract summary: We show that Namioka-Phelps products of compact convex sets agree precisely when one of the two C$*$-algebras is commutative.<n>This confirms an old conjecture by Barker in the case where the compact convex sets are state spaces of C$*$-algebras.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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).

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