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.
Let be unital -algebras faithfully represented on the Hilbert spaces . Let be the tensor product of these Hilbert spaces,
The set of operators of finite sums of form a -subalgebra of . The norm closure of this set is the spatial tensor product of the given -algbras.
Remark: The spatial tensor product does not depend on the chosen faithful representations, see references.
states extend to the spatial tensor product
Let be states on the unitary -algebras. Then there is a unique state on the spatial tensor product such that
Appendix T in the book