Maximal Tensor Product in Operator Systems

• Maximal tensor product is the universal construction assigning the largest matricial positivity structure to operator systems, ensuring all simple tensors of positives are preserved.
• It is defined via Archimedeanization of matrix cones and exhibits symmetry, projectivity, and crucial mapping properties for jointly completely positive maps.
• This construction plays key roles in quantum information and noncommutative geometry by influencing separability, nuclearity, and the factorization of completely positive maps.

The maximal tensor product is a central construction in the theory of operator systems, operator algebras, local operator systems, and related quantum structures. At its core, the maximal tensor product formalizes the "largest" matricial positivity structure compatible with ordered vector space tensoring, exhibiting universal properties with respect to jointly completely positive bilinear maps and factoring behaviors. Its many equivalent characterizations, universal mapping properties, and connections to nuclearity, quotient/projective structures, and twisted quantum group tensor products make it an essential tool across operator algebras, quantum information, and noncommutative geometry.

1. Definition and Fundamental Properties

Let $S$ and $T$ be operator systems, i.e., unital self-adjoint subspaces of $B(H)$ for some Hilbert space $H$. The maximal tensor product $S \otimes_{\max} T$ is the operator-system structure on the algebraic tensor product $S \otimes T$ determined by matrix cones

$$D_{\max}^{n}(S,T) = \left\{\, A (P \otimes Q) A^* \mid P \in M_k(S)^{+}, Q \in M_m(T)^{+}, A \in M_{n, km} \,\right\} \subset M_n(S \otimes T)$$

for each $n$, where $k, m$ vary over $\mathbb{N}$. The final positivity cones are obtained by Archimedeanization: $T$0 The resulting family $T$1 satisfies:

• Each triple $T$2 forms an operator system.
• The construction is symmetric: the swap map $T$3 is a complete order isomorphism.
• The construction is (left and right) projective: it respects operator-system quotients on each factor (Ng, 2015, Han, 2010).

The maximal tensor product achieves the "largest" positive cones making all simple tensors of positives positive, and all jointly completely positive bilinear maps into operator systems factor through it.

2. Universal and Projective Properties

The key universal property is: a bilinear map $T$4 is jointly completely positive if and only if its linearization $T$5 is completely positive. This property uniquely characterizes the maximal tensor product [(Ng, 2015), Thm 2.1].

For operator systems $T$6 and a complete order quotient map $T$7, the induced

$T$8

is a complete order quotient map, confirming that $T$9 is projective, though it is not injective in the category of operator systems. A failure of injectivity distinguishes nuclearity properties such as ($B(H)$0)-nuclearity and their relation to the weak expectation property (WEP) (Han, 2010).

A similar universality and maximality are observed in the context of local operator systems: for any local operator systems $B(H)$1, the maximal ("lmax") tensor product $B(H)$2 is the largest matrix ordering such that every local completely positive bilinear map factors uniquely through it (Beniwal et al., 2020).

3. Equivalent Constructions: Schur Tensor Product and CPAP

The maximal tensor product of operator systems can be characterized equivalently via:

• Schur Tensor Product: For $B(H)$3, $B(H)$4,

$B(H)$5

and $B(H)$6. The maximal and Schur tensor products coincide after Archimedeanization (Ng, 2015).

• Factorization via Completely Positive Approximation Property (CPAP): For finite-dimensional $B(H)$7, a tensor $B(H)$8 corresponds to a map $B(H)$9 with $H$0. Then

$H$1

This provides a factorization characterization of the positive cone of the maximal tensor product [(Ng, 2015), Thm 16–17].

The CPAP (existence of such approximate factorizations for the identity map) is equivalent to $H$2-nuclearity.

4. Maximal Tensor Product in Quantum Group and Twisted Settings

The maximal tensor product also appears in the context of quantum groups and their twisted tensor products. For C*-quantum groups $H$3, $H$4, their coactions $H$5, and a bicharacter $H$6, the maximal $H$7-twisted tensor product $H$8 is defined via universality:

• It is universal with respect to all representations with "braided" commutation relations prescribed by $H$9.
• Every twisted-commutative representation factors uniquely through this maximal product.
• It provides a monoidal structure on categories of coactions and bicategorial horizontal composition for Yetter-Drinfeld C*-algebras (Roy et al., 2015).

In the abelian group case, the maximal twisted tensor product coincides, at the level of C*-envelopes, with the Rieffel deformation by 2-cocycles on the maximal tensor product algebra. The maximal construction surjects onto the minimal twisted tensor product; the two coincide under nuclearity or faithful Drinfeld double coaction (Roy et al., 2015).

5. Maximal Tensor Product for Local Operator Systems

For local operator systems $S \otimes_{\max} T$0, labeled by collections of cones for each index $S \otimes_{\max} T$1, the maximal local tensor product ("lmax") is specified by: $S \otimes_{\max} T$2 the cones are then Archimedeanized. $S \otimes_{\max} T$3 is maximal in the sense that all other local-operator-system tensor products have smaller cones for matching indices. Functoriality, symmetry, associativity, projective limit compatibility, and injectivity for embeddings are preserved. The lmax construction generalizes OMAX in the local setting and recovers it for ordinary (single-cone) operator systems (Beniwal et al., 2020).

6. Further Properties and Applications

• Relation to Minimal Tensor Product: There is always a natural complete order embedding of the minimal into the maximal tensor product; equality holds iff the operator systems are nuclear (Ng, 2015, Beniwal et al., 2020).
• Nuclearity and WEP: ($S \otimes_{\max} T$4)-nuclearity (and variants such as ($S \otimes_{\max} T$5)-nuclearity) relate to the weak expectation property (WEP) for operator systems (Han, 2010).
• Duality: For finite $S \otimes_{\max} T$6, $S \otimes_{\max} T$7, refining the Choi–Effros–Paulsen duality (Ng, 2015).
• Projectivity and Duality for Quotient Maps: Projectivity with respect to complete order quotient maps corresponds, at the dual level, to complete order embeddings under Werner's unitization (Han, 2010).
• Operator Space Projective Norm: For operator systems given by C*-algebras, the maximal tensor product recovers the classical maximal C*-tensor norm/operator space projective tensor norm (Ng, 2015).
• Applications in Quantum Information: In the Hilbert space context, a "maximal" tensor product operator is the universal bilinear map whose universality dictates the structure of multipartite systems; the choice of tensor product influences separability and locality of quantum states and observables (Blair et al., 2021).
• Group-Theoretic and Categorical Interpretations: The maximal tensor product aligns with universal (or "free") constructions in category theory, often realized as coproducts or free objects with respect to positivity structures (Roy et al., 2015, Blair et al., 2021).

7. Summary Table: Definitional Aspects

Context	Maximally Universal Property	Construction/Characterization
Operator systems	All jointly c.p. bilinear maps factor c.p.	Completion of cones $S \otimes_{\max} T$8; Schur tensor; CPAP
Local operator systems	All local c.p. bilinear maps factor l.c.p.	Largest compatible cones across indices; Archimedean.
Quantum group twisted prod.	Universal among braided commutative reps	Universal C*-completion over all braided pairs
Hilbert spaces	All bilinear maps factor through tensor prod.	Uniqueness up to unitary; separability is dependent

The maximal tensor product provides a universal, categorical, and operational framework connecting matrix ordering, quantum symmetries, nuclearity, and the structure of noncommutative tensor products across operator system theory and quantum algebra (Ng, 2015, Roy et al., 2015, Han, 2010, Beniwal et al., 2020, Blair et al., 2021).