Tensor products ofW∗-algebras
Donald Bures · Pacific Journal of Mathematics · 1968
This paper deals primarily with a characterization of the tensor products of a family of ΫP*-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 (J^f) of IF*-algebras is defined to be a T7*-algebra S/ together with injections α* of Stfi into Sf satisfying four conditions: the first two are that the α^J^f) commute and generate S/\ the last two are conditions on the set of positive normal functionals of S/ which are products with respect to the α (J^).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 (S/ϊ) are precisely the incomplete direct products ®{S/i, μi), and that every tensor product is a direct sum of local tensor products which are not product isomorphic.Suppose that (j^J) ίe / is a family of TF*-algebras.We cal tf»)iez) a product for the family (s^) ieI if Szf is a ΫF*-algebra if, for each iel, (Xi is an injection of j^J into jy with a^l) = 1 and if the following conditions hold:(I). a^j& ) commutes with a s (j^) for all i,jel with i φj.(II).^{αr^j^j): i el} = jy: that is, jy is the smallest W* subalgebra of stf which contains all s^Ί for i e I.