C*-tensor norm - Knowlpedia

Let $A$ and $B$ be $C^*$ -algebras. A $C^*$ -tensor norm on their algebraic tensor product $A\odot B$ is a norm $\gamma$ for which the natural multiplication and involution extend to the completion and

\|x^*x\|_\gamma=\|x\|_\gamma^2\qquad(x\in A\odot B).

The completion is denoted $A\otimes_\gamma B$ . Such a norm restricts to the given norms on the factors and satisfies $\|a\otimes b\|_\gamma=\|a\|\,\|b\|$ . Distinct $C^*$ -tensor norms can therefore agree on elementary tensors while assigning different norms to finite sums of them.

Extremal norms

Every $C^*$ -tensor norm lies between two canonical ones:

\|x\|_{\min}\leq \|x\|_\gamma\leq\|x\|_{\max}.

The minimal norm is obtained from faithful spatial representations of the factors. The maximal norm is the supremum over all compatible representations and has a universal property. Thus a $C^*$ -tensor norm amounts to a completion intermediate between the spatial and universal completions.

Why the algebraic tensor product is insufficient

The algebraic tensor product remembers bilinear algebra but carries no distinguished complete norm. For finite-dimensional matrix algebras the $C^*$ -tensor norm is unique, while for general algebras uniqueness can fail. Asking for uniqueness against every second $C^*$ -algebra leads to nuclearity.

For example, representing $A$ on $H$ and $B$ on $K$ gives a norm from the action of $A\odot B$ on $H\otimes K$ . In contrast, representing both factors with commuting ranges on one Hilbert space contributes to the maximal norm. These constructions agree on $a\otimes b$ , but need not agree on a sum $\sum_i a_i\otimes b_i$ .

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV on tensor products of C*-algebras.
Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. DOI record. Relevant: §2.3 on minimal and maximal tensor products and nuclearity.