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

Authors: A. A. Rakhimov; Muzaffar Nurillaev; Zabidin Salleh
Journal: Abstract and Applied Analysis
 (2022)

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## 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 IIIλ (0<λ<1), and a countable number of pairwise nonisomorphic real injective factors of type III0, 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⊂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 “II1”-factors, that is, it is shown that if R is a real factor of type II1, 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′∩R=ℝI where I is the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if R1 is a finite real factor, R2 is a finite real W∗-algebra, and R is a real W∗-subalgebra of R1⊗¯R2 containing R1⊗¯ℝI, then there is some real W∗ -subalgebra Q2⊂R2 such that R=R1⊗¯Q2. Moreover, it is given some affirmative answers to the question of S. Popa for the real case.


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## Body

## 1. Introduction

Injective (complex or real) W∗-algebras with particular factors have been studied well enough. On the other hand, 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, it will be described below that 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, J. Shen, S. Stratila, and L. Zsido [1–4], [5–8], [9]. Now, in more detail, we present the main points of these articles. If N is the algebra (more precisely, factor) of all bounded linear operators acting on a complex Hilbert space H, i.e., if N=BH, and M⊂N is a subfactor, then by the well-known theorem of von Neumann, the bicommutant of M in N is equal to M, i.e., M′′=M (see [8, 9]). But if N is a continuous factor, then in general the bicommutant of M in N is not equal to M (see [4]). If we denote the relative commutants of M in N and BH by MN′ and M′ (=M′BH), respectively, then in general, its relatives are not related, that is, MN′′⊂MBH′′ and MBH′′⊂MN′′. Actually, it seems that the typical and more interesting type of imbedding M as a subfactor of a continuous factor N is such that the bicommutant of M in N is trivial, i.e., M′∩N=ℂI, and in this case M is called irreducible in N. For example, let α:G⟶AutM be a properly outer action of a discrete group G on M. Then M is naturally imbedded in the crossed product algebra M×αG, and by the relative commutant theorem, we have M′∩M×αG=ℂI ([8, 9]). A sufficient condition for a subfactor M of the factor N to have a trivial relative commutant is that there exists an abelian ∗-subalgebra A⊂M which is maximal abelian in N (see [1–3], [4, 6]). At the Baton Rouge (USA) conference in 1967, R.V. Kadison asked the question whether or not the converse is true, namely:IfMis an irreducible subfactor of a factorN, will some maximal abelian∗-subalgebra ofMbe a maximal abelian∗-subalgebra ofN?This problem has been solved in some particular cases. Namely, in 1981, Popa [2] proved that if N is a separable factor, M⊂N is a semifinite irreducible subfactor, and there exists a normal conditional expectation E:N⟶M, then there is an abelian W∗-subalgebra A⊂M which is a maximal abelian W∗-subalgebra in N. Kadison also conjectured that every separable abelian W∗-subalgebra of a factor is contained in some hyperfinite subfactor. Howeover, Kadison’s conjecture turned out to be wrong. In 1983, Popa [1] gave a counterexample in which an abelian W∗-algebra can be imbedded in a factor of type II1 as a maximal injective W∗-subalgebra, i.e., it is not an imbedding as a hyperfinite subfactor. This was also mentioned in [10]. Recall that a W∗-subalgebra M⊂N is called maximal injective in N if it is injective and maximal, i.e., there is no other injective W∗-subalgebra containing M. Note that every W∗-algebra N has one maximal injective W∗-subalgebra at least. Indeed, if Mα is a family of injective W∗-subalgebras of N which is inductively ordered by inclusion, then the weak operator closure of ⋃αMα is an injective W∗-subalgebra of N which contains all Mα. By Zorn’s lemma, N has a maximal injective W∗-subalgebra. We also note that the maximality of an injective W∗-subalgebra and the maximality of a hyperfinite subfactor are not connected with each other, i.e., one does not imply the other. In this line, at first, Murray and von Neumann assumed that any separable factor of type II1 contains a hyperfinite subfactor as a maximal injective W∗-subalgebra. This result was proved by Sorin Popa [1] in 1983. Without the condition of separability, the problem was affirmatively solved by Fang [5].On the other hand, it turns out that the maximality of an injective W∗ -subalgebra is related to its irreducibility. Namely, in 2007, Fang [5] (see also [6]) showed that if M is a maximal hyperfinite subfactor of a factor N, then M is a maximal injective W∗-subalgebra if and only if it is irreducible, i.e., M′∩N=ℂI. Thus, if we look chronologically, the maximality of an injective W∗-subalgebra is concerned with the bicommutant theorem.In this paper, we will investigate the real analogues of the considered notions and the results above. Let us explain the topicality of considering the real analogue of problems in the theory of W∗-algebras. In the mid-1960s, in the works of D. Topping and E. Stormer, Jordan (nonassociative real) analogues of von Neumann algebras JW-algebras were first considered, i.e., real linear spaces of self-adjoint algebraic operators BH, that contain I, are closed under Jordan multiplication a∘b=ab+ba/2 and are closed in the weak operator topology. The structure of these algebras turned out to be close to the structure of von Neumann algebras (that is, W∗-algebras), and in the study of JW-algebras, it turned out to be possible to apply ideas similar to those of the theory of W∗-algebras. In these works, an initial classification of JW-algebras by types was obtained, JW-algebras of type I were completely studied, and it was shown that the study of types II and III is essentially reduced to the study of real W∗-algebras of the corresponding type. Since the mid-1980s, the theory of real W∗-algebras has received significant development. 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<λ<1), and a countable number of pairwise nonisomorphic real injective factors of type III0, whose enveloping 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.


## 2. Preliminaries

LetA be a Banach ∗-algebra over the field ℂ. The algebra A is called a C∗- algebra if aa∗=a2 for any a∈A. A C∗-algebra M is called a W∗-algebra if there exists a Banach space M∗, so-called a predual of M such that M∗∗=M. Let BH be the algebra of all bounded linear operators acting on a complex Hilbert space H and let M⊂BH be a ∗-subalgebra. The subset M′=a∈BH:ba=ab,∀b∈M is called the commutant of M. It is easy to see that
(1)M⊂M′′=MIV=MVI=⋯andM′=MIII=MV=⋯where M′′=M′′. If M=M′′, then it is called a von Neumann algebra. The set ZM=a∈M:ba=ab,∀b∈M is called the center of M. It is obvious that ZM=M∩M′. A von Neumann algebra M is called a factor if its center is trivial, i.e., ZM=λI:λ∈ℂ, where I is the unit of M. By von Neumann’s bicommutant theorem, any von Neumann algebra is W∗- algebra, i.e., M is weakly closed with I∈M. The converse is also true. Therefore, W∗-algebras are also called von Neumann algebras (see [11]).Now, a real∗-subalgebra R⊂BH with unit I is called a real W∗- algebra if it is weakly closed and R∩iR=0. The smallest (complex) W∗-algebra UR containing R coincides with its complexication R+iR, i.e., UR=R+iR. It is known that R generates a natural involutive (i.e., of order 2) ∗-antiautomorphism αR of UR, namely, αRx+iy=x∗+iy∗, where x+iy∈UR, x,y∈R. In this case R=x∈UR:αx=x∗. Conversely, given a W∗-algebra U and any involutive ∗-antiautomorphism α on U, the set z∈U:αz=z∗ is a real W∗-algebra.LetR⊂BH be a real W∗-algebra. It is known that there is a real Hilbert space Hr with
(2)Hr+iHr=H,R⊂BHr⊂BHr+iBHr=BH.The commutant of a∗-algebra R is defined similarly to the complex case:
(3)R′=a∈BHr:ab=ba,∀b∈R.It is directly verified thatR+iR′=R′+iR′ (see [11]).A real Banach∗-algebra R is called a real C∗- algebra if ∥aa∗∥=∥a∥2 and the element I+aa∗ is invertible for any a∈R. Examples of such algebras are the algebra of all linear bounded operators acting on a real Hilbert space Hr, and the algebra of all compact operators on Hr is a real C∗-algebra, but is not a real W∗-algebra. It is known that R is a real C∗-algebra if and only if the norm on R can be extended to the complexification A=R+iR of R so that A is a C∗-algebra (see [12], 5.1.1). Let A be a C∗-algebra. Denote by MnA the algebra of all n×n matrices over A which is also a C∗-algebra. Recall that a continuous linear mapping φ between two C∗-algebras A and B is called completely positive if for any n≥1, the natural map φn from the C∗-algebra A⊗Mn to the C∗-algebra B⊗Mn defined by
(4)φnai,ji,j=1n=φai,ji,j=1nis positive, where Mn is the C∗-algebra of n×n matrices over ℂ. We say that a W∗-algebra A is injective if the following condition is held: for every C∗-algebra B, for every self-adjoint linear subspace S of B containing the identity I, and for every completely positive linear map φ:S⟶A, there is a completely positive linear map φ¯:B⟶A such that φ¯S=φ ([13, 14]). All the notions above are defined similarly for real C∗- and W∗-algebras. Recall [14] that a W∗-algebra A⊂BH is injective if and only if it has the extension property property of E in short (extension property), i.e., there exists a projection P:BH⟶A such that P=1, PI=I. In this case, the map P is completely positive (see [14]).


## 3. Maximal Injective Real W∗ -Subalgebras

LetQ and R be real W∗-algebras. Similarly to the complex case, we define: a real W∗-subalgebra Q⊂R is called maximal injective in R if it is injective and maximal, i.e. there is no other injective real W∗-subalgebra of R containing Q.Theorem 1.
LetQ be a real W∗-subalgebra of a real W∗-algebra R. If Q+iQ is maximal injective in R+iR, then Q is also maximal injective in R.Proof.
LetQ+iQ be a maximal injective W∗-subalgebra of R+iR. Then by Theorem 8 in [15], Q is injective. If there exists an injective real W∗-subalgebra Q1 of R containing Q, then by Theorem 8 in [15], the algebra Q1+iQ1 is injective and Q+iQ⊂Q1+iQ1⊂R+iR which contradicts the maximality of the injective subalgebra Q+iQ.In general, the converse of Theorem1 is still open, but we will prove in a particular case.Theorem 2.
LetR be a real factor of type II1. If Q is a maximal injective real subfactor in R, then Q+iQ is also a maximal injective subfactor in R+iR.Proof.
LetN⊂R+iR be an injective subfactor with Q+iQ⊂N. Since the class of injective factors of type II1 is unique, then the subfactors Q+iQ and N are isomorphic, i.e., there is an isomorphism θ:N⟶Q+iQ. Let α be involutive ∗-antiautomorphism of Q+iQ, generating Q, i.e., Q=a:αa=a∗. Put β=θ−1αθ. It is clear that β is an involutive ∗-antiautomorphism on N. Put Q0=x∈N:βx=x∗. Then N=Q0+iQ0 and Q0 is a real injective subfactor of N. And, Q′=y:θβθ−1y=y∗⊂R+iR is also injective. But, since Q′⊂Q+iQ and Q is maximal injective, then Q′=Q, consequently, N=Q+iQ. Thus, Q+iQ is a maximal injective subfactor in R+iR.
Now, we prove one auxiliary result.Lemma 3.
LetR be a real W∗-algebra and let Q be a maximal injective real W∗-subalgebra of R. Then Q is singular in R, i.e., for a unitary element u∈R with uQu∗=Q, it follows that u∈Q. In particular Q′∩Q=Q′∩R=Z, where Z is the center of Q.Proof.
Letu be a unitary element in R with uQu∗=Q. Then the real W∗-subalgebra of R generated by Q and the element u is injective. Since Q is maximal injective in R, it follows that u∈Q. In particular, it is held that Q′∩R⊂Q and Z⊂Q′∩R⊂Q′∩Q=Z.
Below we prove the real analogue of the theorem of J. Fang which is the solution to the problem of Murray and von Neumann for real factors.Theorem 4.
LetR be a real factor and let Q be a maximal injective real subfactor of R. Then Q is a maximal injective real W∗-subalgebra in R if and only if Q is irreducible in R, i.e. Q′∩R=ℝI.Proof.
IfQ is a maximal injective real W∗-subalgebra of R, then by Lemma 3, ZQ=Q′∩R. Since Q is a real factor, ZQ=ℝI, i.e., Q′∩R=ℝI. Conversely, suppose that Q′∩R=ℝI. For any injective real W∗-algebra F with Q⊂F⊂R, we have F′∩F=Q′∩R=ℝI. Therefore, F is an injective real subfactor of R. Since Q is a maximal injective real subfactor of R, F=Q.Recall that a positive linear mapping fromR onto Q is called a conditional expectation if Eabc=aEbc for all a,c∈Q, b∈R.Lemma 5.
LetR be a real W∗-algebra and let F,Q be real W∗-subalgebras of R such that F⊂Q. If E is a conditional expectation from R onto Q, then E induces a conditional expectation from F′∩R onto F′∩Q.Proof.
Letx∈F′∩R and y∈F. Then xy=yx. Apply the conditional expectation E to the both sides of the equality xy=yx. Since ⊂QExy=yEx. Thus, Ex∈F′∩Q. Since F′∩Q⊂F′∩R, E is a conditional expectation from F′∩R onto F′∩Q when E is restricted on F′∩R.Lemma 6.
LetFi be a real W∗-algebra and let Si be a real W∗-subalgebra of Fii=1,2. If F is a real W∗-algebra such that S1⊗¯S2⊂F⊂F1⊗¯F2 and there is a conditional expectation E:F1⊗¯F2⟶F, then E induces a conditional expectation from S1′∩F1⊗¯F2 onto S1′∩F1⊗¯F2∩F.Proof.
By Lemma 1.7 in [5], we have that
(5)UF1⊗¯UF2∩US1⊗¯ℂI′=UF1⊗¯UF2∩US1′⊗¯BK=US1′∩UF1⊗¯UF2,where UFi is the enveloping W∗ -algebra of the real W∗ -algebra Fi. In general, if F1 and F2 are real W∗ -algebras, then it holds
(6)UF1⊗¯F2=UF1⊗¯UF2.
ThenF1⊗¯F2∩S1⊗¯ℝI′=S1′∩F1⊗¯F2. By Lemma 5, E induces a conditional expectation from S1′∩F1⊗¯F2 onto S1′∩F1⊗¯F2∩F.From Lemma6, we have the following:Corollary 7.
Let the condition of Lemma6 be satisfied and S1′∩F1=ℝI. Consider the set L=x∈F2:I⊗x∈F. Then E induces a conditional expectation from F2 onto L.Using Lemma6 and Corollary 7, we will prove the real analogue of Ge-Kadison’s theorem, so-called “splitting theorem”.Theorem 8.
IfR1 is a finite real factor, R2 is a finite real W∗-algebra, and R is a real W∗-subalgebra of R1⊗¯R2 containing R1⊗¯ℝI, then there is some real W∗-subalgebra Q2⊂R2 such that R=R1⊗¯Q2.Proof.
LetQ be a real W∗-algebra such that R1⊗¯ℝI⊂Q⊂R1⊗¯R2. Since R1 and R2 are finite, there is a conditional normal expectation E from R1⊗¯R2 onto Q. By Corollary 7, E induces a conditional expectation from R2 onto Q2=T∈R2:I⊗T∈Q which we denote by E2. Then we have Ex⊗y=x⊗E2y∈R1⊗¯Q2, for any x∈R1, y∈R2. Since E is normal, it follows that Q=ER1⊗¯R2⊂R1⊗¯Q2. Since Q⊃R1⊗¯Q2, Q=R1⊗¯Q2.


## 4. Maximal Injective Real W∗-Subalgebras of Tensor Products of Real W∗-Algebras

LetR⊂BH be a real W∗-algebra. It is obvious that if Q is a maximal injective real W∗ -subalgebra of R, then Q′ is the minimal injective real W∗-subalgebra containing the algebra R′. Therefore, for any such algebra, there always exists as a maximal injective real W∗-subalgebra and such a minimal injective W∗-extension.Let us prove one auxiliary result.Lemma 9.
LetR be an abelian real W∗ -algebra, Q be a real W∗ -algebra, and let F2 be a minimal separable injective real W∗ -algebra containing Q. If F is an injective real W∗ -algebra such that
(7)R⊗¯Q⊂F⊂R⊗¯F2,then F=R⊗¯F2.Proof.
By formulas (6) and (7), we have
(8)UR⊗¯UQ⊂UF⊂UR⊗¯UF2.
By Lemma 2.4 from [5], we have UF=UR⊗¯UF2. Since
(9)UR⊗¯UF2=UR⊗¯F2,Q⊂F⊂F2,then from UF=UR⊗¯F2, we get F=R⊗¯F2.The Lemma9 can be proved in another way. Indeed:Proof.
LetR be an abelian real W∗-algebra and let F2 be a minimal separable injective real W∗-extension of a real W∗-algebra Q. Then it is directly verified that UR is also abelian and UF2 is the minimal separable injective W∗ -extension of UQ. By (7) we have
(10)UR⊗¯Q⊂UF⊂UR⊗¯F2.
Then by Lemma 2.4 in [5], we have UF=UR⊗¯F2. Consequently, since UF=UR⊗¯F2 and F⊂R⊗¯F2, it follows that F=R⊗¯F2.Now, we will prove the main result of this section.Theorem 10.
LetR1 be an injective real W∗-algebra and R2 be a separable real W∗-algebra. If Q2 is a maximal injective real W∗-subalgebra of R2, then R1⊗¯Q2 is a maximal injective real W∗-subalgebra of R1⊗¯R2.Proof.
Without loss of generality, we can assume thatR1 and R2 are real W∗-algebras acting on complex Hilbert spaces H and K, respectively. Then by the hypothesis of the theorem, K is separable. Let A be the center of R1. Suppose that Q is an injective real W∗-algebra such that R1⊗¯Q2⊂Q⊂R1⊗¯R2. Then we have R1′⊗¯Q2′⊃Q′⊃R1′⊗¯R2′. Since Q′ is an injective real W∗ -subalgebra of R1′⊗¯Q2′, there is a conditional expectation E from R1′⊗¯Q2′ onto Q′. By Lemma 5, E induces a conditional expectation from A⊗¯Q2′ onto F≔A⊗¯Q2′∩Q′. Consequently, F is an injective real W∗-algebra such that A⊗¯Q2′⊃F⊃A⊗¯R2′. Since Q2 is a maximal injective real W∗-subalgebra of R2, Q2′ is the minimal injective real W ∗ -extension of R2′. By Lemma 9, we get F=A⊗¯Q2′. Thus ℝI⊗¯Q2′⊂F⊂Q′. So, Q′=R1′⊗¯Q2′ and Q=R1⊗¯Q2.


## 5. Some Particular Cases

Let us consider the following problem which was formulated by Popa (see [1]) in the complex case:IfR1andR2are real W∗-algebras,Q1andQ2are maximal injective real W∗-subalgebras ofR1andR2, respectively, then will the algebraQ1⊗¯Q2be a maximal injective real W∗-subalgebra ofR1⊗¯R2?In some certain cases, we will give affirmative answers to the question.(1)
The case whenQ1 is a real factor and R1 is separableAs examples of such algebras, one can takeQ1 as an infinite tensor product of a matrix algebra M2ℝ, which is a real factor of type II1, and R1 as an arbitrary real W∗ -algebra acting on a separable real Hilbert space.Theorem 11.
LetRi⊂BHi be a real W∗-algebra and let Qi be a maximal injective real W∗-subalgebra of Rii=1,2. If R1 is separable and Q1 is a real factor, then Q1⊗¯Q2 is a maximal injective real W∗-subalgebra of R1⊗¯R2.Proof.
LetQ be an injective real W∗-subalgebra with Q1⊗¯Q2⊂Q⊂R1⊗¯R2. Then we obtain Q1′⊗¯Q2′⊃Q′⊃R1′⊗¯R2′. Since Qi is a maximal injective real W∗-subalgebra of Ri, Q′ is the minimal injective real W∗-extension of Ri′i=1,2. Since Q′ is an injective real W∗-subalgebra of the W∗-subalgebra Q1′⊗¯Q2′, there is a conditional expectation E:Q1′⊗¯Q2′⟶Q′.LetL=x∈Q2′:I⊗x∈Q′. Then L⊂Q2′. By Lemma 3 we have
(11)R1′′∩Q1′=Q1∩R1=Q1∩Q1=ℝI.By Corollary7, the mapping E induces a conditional expectation from Q2′ onto L. Thus, L is injective. Since R2′⊂L⊂Q2′ and Q2′ is the minimal injective real W∗-extension of R2′, L=Q2′. So, ℝI⊗¯Q2′⊂Q′. This implies that Q1′⊗¯Q2′⊃Q′⊃R1′⊗¯Q2′ and hence, Q1⊗¯Q2⊂Q⊂R1⊗¯Q2. By Theorem 10, we have Q=Q1⊗¯Q2.
(2)
The case whenR1 is separable and the center of Q1 is atomicRecall that a real or complex von Neumann algebraA is called atomic if every nonzero projection of A majorizes a nonzero minimal projection (atom), i.e., under each nonzero projection, there is an atom.As examples of such algebras, one can takeR1 as an arbitrary real W∗-algebra acting on a separable real Hilbert space, and Q1 as an arbitrary real W∗-algebra whose center is a commutative algebra.Let us prove one auxiliary result which in itself is interesting.Theorem 12.
LetQ be a real W∗-algebra and R be the minimal injective real W∗-algebra containing Q. If θ is a ∗-automorphism of R such that θx=x, ∀x∈Q, then θy=y. That is, if a ∗-automorphism is identical to a real W∗ algebra, then it is also identical to its minimal injective real W∗-extension.Proof.
LetR act on a Hilbert space H, i.e., R⊂BH. θ¯ denotes a linear extension of the ∗-automorphism θ on the W∗-algebra M=R+iR that is defined as θ¯a+ib=θa+iθb. As is known, every ∗-automorphism of BH is inner, i.e., there exists some unitary element u∈BH with θ¯x=uxu∗=Adux, x∈M. By the condition of the theorem, we have uxu∗=x for all x∈Q, consequently, for all x∈N=Q+iQ. Hence, we can get: u∈N′. Put θ′¯yθ′¯y′=uy′u∗, for y′∈M′. It is easy to see that θ′¯y′θ¯y=θ¯yθ′¯y′, for all y∈M, y′∈M′. Since θ¯M=M, θ′¯y′∈M′ for all y′∈M′, i.e., uM′u∗⊂M′. Note that θ¯−1y=u∗yu is also a ∗-automorphism on M. Similar reasoning for the ∗-automorphism θ¯−1, we can obtain u∗M′u⊂M′. So, uM′u∗=M′. It implies that u∈N′ is a normalizer of M′. By Lemma 1.1 in [5], u∈M′. Consequently, θ¯y=uyu∗=y for all y∈M, i.e., θ¯M=id. In particular, θR=id.
Now, using Theorem12, we prove the following auxiliary results.Lemma 13.
IfR is the minimal injective real W∗-extension of a real W∗-algebra Q and p,q are nonzero central orthogonal projections in R, then there is not a ∗-isomorphism φ:Rp⟶Rq with φpx=qx, ∀x∈Q.Proof.
Suppose that there is a∗-isomorphism φ:Rp⟶Rq with φpx=qx, ∀x∈Q. Then y=pq+qy+I−p−qy for all y∈R. Define the mapping θ:R⟶R as
(12)θy=φpy+φ−1qy+I−p−qy.
Sincep,q are mutually orthogonal central projections and the mapping φ:Rp⟶Rq is a ∗-isomorphism, it is easy to see that the mapping θ is a ∗-automorphism of R. Moreover, for any x∈Q, we have also
(13)θx=φpx+φ−1qx+1−p−qx=qx+px+1−p−qx=x.
That is, the∗-automorphism θ acts identically on Q. Then by Theorem 12, we have θy=y, for any y∈R. That is, the ∗-automorphism θ is identical to R. Consequently, p=θp=φp=q. Hence, we obtain that p=pq=0. It contradicts to the assumption that p≠0.Lemma 14.
LetR be the minimal injective real W∗-extension of a real W∗-algebra Q and let p,q be nonzero central projections in R. If there is a ∗-isomorphism φ:Rp⟶Rq with φpx=qx, ∀x∈Q, then p=q and φpy=py for any y∈R.Proof.
Suppose thatp≠q. Let e=pq. Without loss of generality, we can assume that p1=p−e>0. Let q1=θp1≤q. Then p1,q1 are nonzero central projections in R and p1q1=q1p1=0. It is easy to see that the ∗-isomorphism φ:Rp⟶Rq induces a ∗-isomorphism ψ:Rp1⟶Rq1 with ψp1y=φp1y for all y∈R. Moreover, for any x∈Q, we have:
(14)ψp1x=φp1x=φp1φpx=q1qx=q1x.
As we see that it contradicts Lemma13. Thus p=q. Now, define a mapping θ on R as:
(15)θy=ϕpy+1−py.
It is directly shown thatθ is ∗-automorphism of R and θx=x, for all x∈Q. Then by Theorem 12, we have: θy=y for all y∈R. Hence, py=θpy=φpy for all y∈R.
Now, let us prove the main result of the section of the paper.Theorem 15.
LetRi⊂BHi be a real W∗-algebra and let Qi be a maximal injective real W∗-subalgebra of Rii=1,2. If R1 is separable and the center of Q1 is atomic, then Q1⊗¯Q2 is maximal injective in R1⊗¯R2.Proof.
LetQ be an injective real W∗-subalgebra with
(16)Q1⊗¯Q2⊂Q⊂R1⊗¯R2.
To prove the theorem, we have to show thatQ=Q1⊗¯Q2. By Theorem 10, it is sufficient to prove that Q⊆R1⊗¯Q2. For that, it is enough to prove that ℝI⊗¯Q2′⊂Q′. Put A=Q1∩Q1′=R1∩Q1′, i.e., let A be the center of Q1 which is atomic according to the condition. Since the algebras Q1⊂R1 are separable, then the set of minimal projections is countable, which we denote by pn:n=1,2,⋯. Put F=Q′∩A⊗¯Q2′=Q′∩R1′⊗¯ℝI′. As in the proof of Theorem 10, it is similarly shown that F is injective and ℝI⊗¯R2′⊆F⊆A⊗¯Q2′.Note thatpn⊗I∈F′ for every n. Since Q2′ is the minimal injective real W∗-extension of R2, we have Fpn⊗I=pn⊗Q2′ for every n. We denote by zn the smallest projection in ZF satisfying the condition: pn⊗I≤zn. Then we can get a ∗-isomorphism θn:Q2′⟶Fzn uniquely determined by the formula
(17)θnypn⊗I=pn⊗yforally∈Q2′.SinceℝI⊗¯R2′⊆F, it follows that θnx=I⊗xzn for all x∈R2′.Thus for alln,m and for all x∈R2′, we have θnxzm=θmxzn. Since Q2′ is the minimal injective real W∗-extension of R2′, then by Lemma 14, the last equation is satisfied for all y∈Q2′ and all n,m. Moreover, hence, for every y∈Q2′ there is an element a∈F such that azn=θny for all n. In particular apn⊗I=pn⊗y, i.e., a=I⊗y. Therefore, ℝI⊗¯Q2′⊆F.
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The case whenQi, Rii=1,2 are real factorsTheorem 16.
LetRi⊂BHi be a real factor and Qi be a maximal injective real subfactor of Ri, i=1,2. If Q1′∩R1≅ℝN1≤N≤∞ and Q2′∩R2=ℝI, then Q1⊗¯Q2 is a maximal injective real subfactor of R1⊗¯R2.Proof.
LetQ be an injective real W∗-algebra with
(18)Q1⊗¯Q2⊂Q⊂R1⊗¯R2.
Then we haveQ1′⊗¯Q2′⊃Q′⊃R1′⊗¯R2′. By Theorem 4, Q2 is a maximal injective real W∗-subalgebra of R2, and consequently, Q2′ is the minimal injective real W∗-extension of R2′. By assumption, Q2′∩R2=Q2′∩Q2=ℝI. Let E:Q1⊗¯Q2⟶Q′ be a conditional expectation and F=Q1′∩R1≅ℝN. By Lemma 6, E induces a conditional expectation from F⊗¯Q2′ onto B, where B=F⊗¯Q2′∩Q′. Therefore, B is an injective real W∗-algebra such that
(19)Q′⊇B⊇F⊗¯Q2′∩R1′⊗¯R2′=ℝI⊗¯R2′.
Similar arguments as the proof of Theorem15 shows that B⊇ℝI⊗¯Q2′. So, Q′⊇ℝI⊗¯Q2′. By Theorem 8 (the real analogue of Ge-Kadison’s splitting theorem), for some real W∗ -subalgebra N′⊆Q1′ we have: Q′=N′⊗¯Q2′. Therefore, Q=N⊗¯Q2. Since Q is injective, it follows that N is also injective and Q1⊆N⊆R1. Since Q1 is a maximal injective real subfactor, then N=Q1. Consequently, Q=Q1⊗¯Q2.Corollary 17.
LetRi be a real factor, Qi be an irreducible, maximal injective real subfactor of Rii=1,2. Then Q1⊗¯Q2 is an irreducible, maximal injective real subfactor of R1⊗¯R2. Moreover, by Theorem 4, Q1⊗¯Q2 is a maximal injective real W∗ -subalgebra of the factor R1⊗¯R2.


## 6. Conclusion

In the article, the maximal injective real W∗-subalgebras of real W∗-algebras or real factors are investigated. For real factors Q⊂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 may not be true. It is assumed that using the crossed product of a finite real W∗-algebra on an action, one can construct a maximal injective real subfactor of type III, for which the enveloping complex subfactor is not the maximal injective. Now the authors of the article are preparing a continuation of this article, where this example will be given. Nevertheless, in this article, the validity of the converse statement for finite factors is proved. Exactly, the converse is proved in the case “II1”-factors, that is, it is shown that if R is a real factor of type II1, 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. The above show how the real case differs from the complex case, and this is the contribution of this article. Moreover, the “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if R1 is a finite real factor, R2 is a finite real W∗-algebra, and R is a real W∗-subalgebra of R1⊗¯R2 containing R1⊗¯ℝI, then there is some real W∗-subalgebra Q2⊂R2 such that R=R1⊗¯Q2. And also, it is given some affirmative answers to the question of S. Popa for the real case.



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*Source: 4039408-2022-04-25.xml*