This paper deals primarily
with a characterization of the tensor products of a family of W∗-algebras
(abstract von Neumann algebras). It is especially concerned with infinite
tensor products; the results, however, apply and have interest in the finite
case.
A tensor product for a family (𝒜i) of W∗-algebras is defined to be a W∗-algebra
𝒜 together with injections αi of 𝒜i into 𝒜 satisfying four conditions: the first two are
that the αi(𝒜i) commute and generate 𝒜; the last two are conditions on the set of
positive normal functionals of 𝒜 which are products with respect to the αi(𝒜i).
A local tensor product is defined to be a tensor product satisfying a fifth
condition—that its tail reduce to the scalars. It is shown that the local tensor
products of (𝒜i) are precisely the incomplete direct products ⊗(𝒜i,μi), and that
every tensor product is a direct sum of local tensor products which are not product
isomorphic.
