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

Abstract

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 ${\mathrm{I}\mathrm{I}\mathrm{I}}_{\lambda }(0<\lambda <1)$, and a countable number of pairwise nonisomorphic real injective factors of type ${\text{III}}_0$, whose enveloping (complex) $W*$-factors are isomorphic, is constructed. It follows from the above that the study of the real analogue of problems in the theory of operator algebras is topical. Moreover, the real analogue is a generalization of the complex case, since the class of real linear operators is much wider than the class of complex linear operators. In this paper, the maximal injective real $W*$-subalgebras of real $W*$-algebras or real factors are investigated. For real factors $Q\subset R$, it is proven that if $Q+iQ$ is a maximal injective $W*$-subalgebra in $R+iR$, then $Q$ also is a maximal injective real $W*$-subalgebra in $R$. The converse is proved in the case “ ${\text{II}}_1$”-factors, that is, it is shown that if $R$ is a real factor of type ${\text{II}}_1$, then the maximal injectivity of $Q$ implies the maximal injectivity of $Q+iQ$. Moreover, it is proven that a maximal injective real subfactor $Q$ of a real factor $R$ is a maximal injective real $W*$-subalgebra in $R$ if and only if $Q$ is irreducible in $R$, i.e., $Q\prime \cap R=ℝ𝕀$ where $𝕀$ is the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if $R_1$ is a finite real factor, $R_2$ is a finite real $W*$-algebra, and $R$ is a real $W*$-subalgebra of $R_1\overline{\otimes }{R}_2$ containing $R_1\overline{\otimes }ℝ𝕀$, then there is some real $W*$-subalgebra $Q_2\subset R_2$ such that $R=R_1\overline{\otimes }{Q}_2$. Moreover, it is given some affirmative answers to the question of S. Popa for the real case.