Entanglement in C^*-algebras: tensor products of state spaces · arXivDesk
2512.10410Dec 11, 202527 pages. This version has undergone significant revisions, including correcting a wrong statement about tensor products of Poulsen simplexes
Entanglement in C∗-algebras: tensor products of state spaces
We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C∗-algebras. Relatedly, we study entanglement in (infinite dimensional) C∗-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C∗-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C∗
Nearby in the stack
-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C
∗
-algebras. We show that the minimal and maximal tensor products of state spaces of C
∗
-algebras agree precisely when one of the two C
∗
-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C
∗
-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C
∗
-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C
∗
-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C
∗
-algebras.
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