nLab
spatial tensor product

Contents

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\mathcal{A}_1, ..., \mathcal{A}_k be unital C *C^*-algebras faithfully represented on the Hilbert spaces H 1,...,H kH_1, ..., H_k. Let HH be the tensor product of these Hilbert spaces,

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

The set of operators of finite sums of A 1... kA kA_1 \otimes ... \otimes_k A_k form a **-subalgebra of (H)\mathcal{B}(H). The norm closure of this set is the spatial tensor product of the given C *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 ρ 1,...,ρ k\rho_1, ..., \rho_k be states on the unitary C *C^*-algebras. Then there is a unique state ρ\rho on the spatial tensor product such that

ρ(A 1... kA k)=ρ 1(A 1)ρ 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 *C^*-algebras: a friendly approach. (ZMATH)

Revised on April 10, 2013 21:07:48 by Urs Schreiber (131.174.41.18)