Minimal C*-tensor product - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be $C^*$ -algebras. Choose faithful nondegenerate representations $\pi:A\to B(H)$ and $\rho:B\to B(K)$ . The minimal $C^*$ -tensor norm is

\|x\|_{\min}=\|(\pi\otimes\rho)(x)\|_{B(H\otimes K)} \qquad(x\in A\odot B),

and the completion is the minimal $C^*$ -tensor product $A\otimes_{\min}B$ . The norm is independent of the chosen faithful representations. It is the smallest $C^*$ -tensor norm on $A\odot B$ , which explains both “minimal” and the alternative name “spatial.”

Spatial construction

On elementary tensors the representation is

(\pi\otimes\rho)(a\otimes b)=\pi(a)\otimes\rho(b).

The images act on separate Hilbert-space factors and therefore commute in the appropriate tensor-product sense. Faithfulness of $\pi$ and $\rho$ makes the resulting algebraic representation faithful. The nontrivial independence theorem says that changing either faithful representation leaves the induced norm unchanged Takesaki, Chapter IV, §4.

Functoriality and injectivity

Given $*$ -homomorphisms $\varphi:A\to C$ and $\psi:B\to D$ , the algebraic map $\varphi\odot\psi$ extends to

\varphi\otimes\psi:A\otimes_{\min}B\longrightarrow C\otimes_{\min}D.

If both maps are injective, so is their minimal tensor product. This injectivity property is one reason for the notation $\otimes_{\min}$ and the occasional name “injective tensor product.” It does not assert that minimal tensoring preserves arbitrary short exact sequences; that stronger property is exactness of a $C^*$ -algebra.

Standard models

For a locally compact Hausdorff space $X$ ,

C_0(X)\otimes_{\min}B\cong C_0(X,B),

where the right side consists of norm-continuous $B$ -valued functions vanishing at infinity. Also $M_n(\mathbb C)\otimes_{\min}B\cong M_n(B)$ , the matrix $C^*$ -algebra over $B$ . These examples expose the spatial meaning: scalar functions or matrices acquire coefficients in $B$ without introducing an additional tensor norm.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV, especially §4, on spatial 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 C-tensor products.