C*-tensor norm - Imported Knowlpedia Content

Definition

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 Takesaki, Chapter IV.

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.