nLab
spatial tensor product

Idea

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

$$H:={\otimes }_{i=1}^k{H}_k$$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 $ℬ(H)$. 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

$$\rho (A_1\otimes ...{\otimes }_k{A}_k)={\rho }_1(A_1)\cdots {\rho }_k(A_k)$$\rho(A_1 \otimes ... \otimes_k A_k) = \rho_1(A_1) \cdots \rho_k(A_k)

References

Appendix T in the book

• Niels Erik Wegge-Olsen: K-theory and $C^{*}$-algebras: a friendly approach. (ZMATH)