Maximal Injective Real W∗-Subalgebras of Tensor Products of Real W∗-Algebras

It is known that injective (complex or real) W∗-algebras with particular factors have been studied well enough. In the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) W∗-algebras is hard enough, in particular, there exist continuum pairwise nonisomorphic noninjective factors of type II. Therefore, it seems interesting to study maximal injective W∗-subalgebras and subfactors. On the other hand, the study of maximal injective W∗-subalgebras and subfactors is also related to the well-known von Neumann’s bicommutant theorem. In the complex case, such subalgebras were investigated by S. Popa, L. Ge, R. Kadison, J. Fang, and J. Shen. In recent years, studies have also begun in the real case. Let us briefly recall the relevance of considering the real case. It is known that in the works of D. Topping and E. Stormer, it was shown that the study of JW-algebras (nonassociative real analogues of von Neumann algebras) of types II and III is essentially reduced to the study of real W∗-algebras of the corresponding type. It turned out that the structure of real W∗-algebras, generally speaking, differs essentially in the complex case. For example, in the finite-dimensional case, in addition to complex and real matrix algebras, quaternions also arise, i.e., matrix algebras over quaternions. In the infinite-dimensional case, it is proved that there exist, up to isomorphism, two real injective factors of type IIIλ (0