Maximal C*-tensor product

Let $A$ and $B$ be $C^*$ -algebras. For $x=\sum_i a_i\otimes b_i\in A\odot B$ , define

\|x\|_{\max}=\sup_{\pi_A,\pi_B} \left\|\sum_i\pi_A(a_i)\pi_B(b_i)\right\|,

where the supremum runs over pairs of $*$ -representations on a common Hilbert space whose ranges commute. This finite supremum is the maximal $C^*$ -tensor norm, and the completion $A\otimes_{\max}B$ is the maximal $C^*$ -tensor product. It is the largest $C^*$ -tensor norm on $A\odot B$ .

Universal property

If $\varphi:A\to M(C)$ and $\psi:B\to M(C)$ are nondegenerate $*$ -homomorphisms with commuting ranges, there is a unique $*$ -homomorphism

\varphi\mathbin{\cdot}\psi:A\otimes_{\max}B\longrightarrow M(C)

satisfying $(\varphi\mathbin{\cdot}\psi)(a\otimes b)=\varphi(a)\psi(b)$ . Conversely, representations of $A\otimes_{\max}B$ yield commuting representations of the factors. This is the precise sense in which the maximal tensor product is universal.

Comparison with the minimal product

Because $\|x\|_{\min}\leq\|x\|_{\max}$ , the identity on $A\odot B$ extends to a canonical surjective map

A\otimes_{\max}B\longrightarrow A\otimes_{\min}B.

The target is the minimal $C^*$ -tensor product. The map is injective exactly when the two canonical norms agree for this pair $(A,B)$ . Requiring agreement for every $B$ defines a nuclear $C^*$ -algebra.

For matrix algebras, and more generally when one factor is nuclear, the canonical map is an isomorphism. Without such a hypothesis its kernel may be nonzero, so the notation $A\otimes B$ is ambiguous.

Functoriality and quotients

Every pair of $*$ -homomorphisms $\varphi:A\to C$ and $\psi:B\to D$ induces a $*$ -homomorphism $\varphi\otimes_{\max}\psi:A\otimes_{\max}B\to C\otimes_{\max}D$ . Maximal tensoring also has a strong quotient property: if $A\to A/I$ is the quotient map, then the induced map onto $(A/I)\otimes_{\max}B$ has kernel generated by $I\odot B$ . This behavior contrasts with the injectivity property of the minimal product.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV on universal and spatial C*-tensor products.
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 maximal tensor products and nuclearity.