Maximal Injective Real W<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mo>∗</a:mo></a:math>-Subalgebras of Tensor Products of Real W<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mo>∗</c:mo></c:math>-Algebras
Annotatsiya
It is known that injective (complex or real) W <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M3"><a:mo>∗</a:mo></a:math> -algebras with particular factors have been studied well enough. In the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) W <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M4"><c:mo>∗</c:mo></c:math> -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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M5"><e:mo>∗</e:mo></e:math> -subalgebras and subfactors. On the other hand, the study of maximal injective W <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M6"><g:mo>∗</g:mo></g:math> -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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M7"><i:mo>∗</i:mo></i:math> -algebras of the corresponding type. It turned out that the structure of real W <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M8"><k:mo>∗</k:mo></k:math> -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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M9"><m:msub><m:mrow><m:mtext>III</m:mtext></m:mrow><m:mrow><m:mi>λ</m:mi></m:mrow></m:msub></m:math> ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M10"><o:mn>0</o:mn><o:mo><</o:mo><o:mi>λ</o:mi><o:mo><</o:mo><o:mn>1</o:mn></o:math> ), and a countable number of pairwise nonisomorphic real injective factors of type <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M11"><q:msub><q:mrow><q:mtext>III</q:mtext></q:mrow><q:mrow><q:mn>0</q:mn></q:mrow></q:msub></q:math> , whose enveloping (complex) W <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M12"><s:mo>∗</s:mo></s:math> -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 <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M13"><u:mo>∗</u:mo></u:math> -subalgebras of real W <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M14"><w:mo>∗</w:mo></w:math> -algebras or real factors are investigated. For real factors <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M15"><y:mi>Q</y:mi><y:mo>⊂</y:mo><y:mi>R</y:mi></y:math> , it is proven that if <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M16"><ab:mi>Q</ab:mi><ab:mo>+</ab:mo><ab:mi>i</ab:mi><ab:mi>Q</ab:mi></ab:math> is a maximal injective W <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M17"><cb:mo>∗</cb:mo></cb:math> -subalgebra in <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M18"><eb:mi>R</eb:mi><eb:mo>+</eb:mo><eb:mi>i</eb:mi><eb:mi>R</eb:mi></eb:math> , then <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M19"><gb:mi>Q</gb:mi></gb:math> also is a maximal injective real W <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M20"><ib:mo>∗</ib:mo></ib:math> -subalgebra in <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M21"><kb:mi>R</kb:mi></kb:math> . The converse is proved in the case “ <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M22"><mb:msub><mb:mrow><mb:mtext>II</mb:mtext></mb:mrow><mb:mrow><mb:mn>1</mb:mn></mb:mrow></mb:msub></mb:math> ”-factors, that is, it is shown that if <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M23"><ob:mi>R</ob:mi></ob:math> is a real factor of type <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M24"><qb:msub><qb:mrow><qb:mtext>II</qb:mtext></qb:mrow><qb:mrow><qb:mn>1</qb:mn></qb:mrow></qb:msub></qb:math> , then the maximal injectivity of <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" id="M25"><sb:mi>Q</sb:mi></sb:math> implies the maximal injectivity of <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" id="M26"><ub:mi>Q</ub:mi><ub:mo>+</ub:mo><ub:mi>i</ub:mi><ub:mi>Q</ub:mi></ub:math> . Moreover, it is proven that a maximal injective real subfactor <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" id="M27"><wb:mi>Q</wb:mi></wb:math> of a real factor <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" id="M28"><yb:mi>R</yb:mi></yb:math> is a maximal injective real W <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" id="M29"><ac:mo>∗</ac:mo></ac:math> -subalgebra in <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" id="M30"><cc:mi>R</cc:mi></cc:math> if and only if <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" id="M31"><ec:mi>Q</ec:mi></ec:math> is irreducible in <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" id="M32"><gc:mi>R</gc:mi></gc:math> , i.e., <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" id="M33"><ic:msup><ic:mrow><ic:mi>Q</ic:mi></ic:mrow><ic:mrow><ic:mo>′</ic:mo></ic:mrow></ic:msup><ic:mo>∩</ic:mo><ic:mi>R</ic:mi><ic:mo>=</ic:mo><ic:mi>ℝ</ic:mi><ic:mi mathvariant="double-struck">I</ic:mi></ic:math> where <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" id="M34"><lc:mi mathvariant="double-struck">I</lc:mi></lc:math> is the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" id="M35"><oc:msub><oc:mrow><oc:mi>R</oc:mi></oc:mrow><oc:mrow><oc:mn>1</oc:mn></oc:mrow></oc:msub></oc:math> is a finite real factor, <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" id="M36"><qc:msub><qc:mrow><qc:mi>R</qc:mi></qc:mrow><qc:mrow><qc:mn>2</qc:mn></qc:mrow></qc:msub></qc:math> is a finite real W <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" id="M37"><sc:mo>∗</sc:mo></sc:math> -algebra, and <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" id="M38"><uc:mi>R</uc:mi></uc:math> is a real W <wc:math xmlns:wc="http://www.w3.org/1998/Math/MathML" id="M39"><wc:mo>∗</wc:mo></wc:math> -subalgebra of <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" id="M40"><yc:msub><yc:mrow><yc:mi>R</yc:mi></yc:mrow><yc:mrow><yc:mn>1</yc:mn></yc:mrow></yc:msub><yc:mover accent="false"><yc:mrow><yc:mo>⊗</yc:mo></yc:mrow><yc:mrow><yc:mo>¯</yc:mo></yc:mrow></yc:mover><yc:msub><yc:mrow><yc:mi>R</yc:mi></yc:mrow><yc:mrow><yc:mn>2</yc:mn></yc:mrow></yc:msub></yc:math> containing <bd:math xmlns:bd="http://www.w3.org/1998/Math/MathML" id="M41"><bd:msub><bd:mrow><bd:mi>R</bd:mi></bd:mrow><bd:mrow><bd:mn>1</bd:mn></bd:mrow></bd:msub><bd:mover accent="false"><bd:mrow><bd:mo>⊗</bd:mo></bd:mrow><bd:mrow><bd:mo>¯</bd:mo></bd:mrow></bd:mover><bd:mi>ℝ</bd:mi><bd:mi mathvariant="double-struck">I</bd:mi></bd:math> , then there is some real W <fd:math xmlns:fd="http://www.w3.org/1998/Math/MathML" id="M42"><fd:mo>∗</fd:mo></fd:math> -subalgebra <hd:math xmlns:hd="http://www.w3.org/1998/Math/MathML" id="M43"><hd:msub><hd:mrow><hd:mi>Q</hd:mi></hd:mrow><hd:mrow><hd:mn>2</hd:mn></hd:mrow></hd:msub><hd:mo>⊂</hd:mo><hd:msub><hd:mrow><hd:mi>R</hd:mi></hd:mrow><hd:mrow><hd:mn>2</hd:mn></hd:mrow></hd:msub></hd:math> such that <jd:math xmlns:jd="http://www.w3.org/1998/Math/MathML" id="M44"><jd:mi>R</jd:mi><jd:mo>=</jd:mo><jd:msub><jd:mrow><jd:mi>R</jd:mi></jd:mrow><jd:mrow><jd:mn>1</jd:mn></jd:mrow></jd:msub><jd:mover accent="false"><jd:mrow><jd:mo>⊗</jd:mo></jd:mrow><jd:mrow><jd:mo>¯</jd:mo></jd:mrow></jd:mover><jd:msub><jd:mrow><jd:mi>Q</jd:m