There are several different concepts of tensor products for C-star algebras, because there are different norms one can put on the algebraic tensor product that turns it into a C-star algebra. The spatial tensor product uses the smallest norm of all possible norms. There is also a maximal norm and it is a nontrivial theorem that all norms fall in between these two.

Definition

Let $\mathcal{A}_1, ..., \mathcal{A}_k$ be unital $C^*$-algebras faithfully represented on the Hilbert spaces$H_1, ..., H_k$. Let $H$ be the tensor product of these Hilbert spaces,

$H := \otimes_{i=1}^k H_k$

The set of operators of finite sums of $A_1 \otimes ... \otimes_k A_k$ form a $*$-subalgebra of $\mathcal{B}(H)$. The norm closure of this set is the spatial tensor product of the given $C^*$-algbras.

Remark: The spatial tensor product does not depend on the chosen faithful representations, see references.

Properties

Theorem

states extend to the spatial tensor product

Let $\rho_1, ..., \rho_k$ be states on the unitary $C^*$-algebras. Then there is a unique state $\rho$ on the spatial tensor product such that