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 $^*$ -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.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$^*$-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.

math.OACross-listedarXiv:2512.10410Dec 12, 2025

THE TENSOR PRODUCT OF TENSOR OPERATORS OVER QUANTUM ALGEBRAS: SOME APPLICATIONS TO QUANTUM SPIN CHAINS

Manfred Scheunert

WORLD SCIENTIFIC eBooksJan 1, 19950 citations

Auxiliary $C^*$ -AlgebrasAuxiliary $$C^*$$ -Algebras

Emil V. Prodan

SpringerBriefs in mathematical physicsJan 1, 20170 citations

Coupling capacity in C*-algebras

Adam Skalski, Ivan G. Todorov, Lyudmila B. Turowska

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

arXiv:2211.02401Proceedings of the Royal Society of Edinburgh Section A MathematicsSep 7, 20231 citation

Separable states and positive maps II

Erling Størmer

Using the natural duality between linear functionals on tensor products of $C^*$ -algebras with the trace class operators on a Hilbert space $H$ and linear maps of the $C^*$ -algebra into $B(H)$ , we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF $C^*$ -algebras.Using the natural duality between linear functionals on tensor products of $C^*$-algebras with the trace class operators on a Hilbert space $H$ and linear maps of the $C^*$-algebra into $B(H)$, we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF $C^*$-algebras.

MATHEMATICA SCANDINAVICADec 1, 200915 citations

Tensor products of Hilbert space effect algebras

Sylvia Pulmannová

Reports on Mathematical PhysicsApr 1, 200412 citations

Operator product states on tensor powers of

C^*

-algebras

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

^*

-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.

Auxiliary

C^*

-Algebras

Using the natural duality between linear functionals on tensor products of

C^*

-algebras with the trace class operators on a Hilbert space

H

and linear maps of the

C^*

-algebra into

B(H)

, we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF

C^*

-algebras.