Surjective separating maps on noncommutative L p -spaces
aa r X i v : . [ m a t h . OA ] S e p SURJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES CHRISTIAN LE MERDY AND SAFOURA ZADEH
Abstract.
Let 1 ≤ p < ∞ and let T : L p ( M ) → L p ( N ) be a bounded map betweennoncommutative L p -spaces. If T is bijective and separating (i.e., for any x, y ∈ L p ( M )such that x ∗ y = xy ∗ = 0, we have T ( x ) ∗ T ( y ) = T ( x ) T ( y ) ∗ = 0), we prove the existenceof decompositions M = M ∞ ⊕ M , N = N ∞ ⊕ N and maps T : L p ( M ) → L p ( N ), T : L p ( M ) → L p ( N ), such that T = T + T , T has a direct Yeadon type fac-torisation and T has an anti-direct Yeadon type factorisation. We further show that T − is separating in this case. Next we prove that for any 1 ≤ p < ∞ (resp. any1 ≤ p = 2 < ∞ ), a surjective separating map T : L p ( M ) → L p ( N ) is S -bounded (resp.completely bounded) if and only if there exists a decomposition M = M ∞ ⊕ M suchthat T | L p ( M ) has a direct Yeadon type factorisation and M is subhomogeneous. Introduction
This paper deals with separating maps between noncommutative L p -spaces, 1 ≤ p < ∞ .These operators were investigated recently in [1, 4, 5], to which we refer for background,motivation and historical facts. Recall that a bounded map T : L p ( M ) → L p ( N ) betweentwo noncommutative L p -spaces is called separating if for any x, y ∈ L p ( M ), the condition x ∗ y = xy ∗ = 0 implies that T ( x ) ∗ T ( y ) = T ( x ) T ( y ) ∗ = 0. It was shown in [4, Proposition3.11] and [1, Theorem 3.3 & Remark 3.4] that T : L p ( M ) → L p ( N ) is separating if and onlyif there exists a w ∗ -continuous Jordan homomorphism J : M → N , a positive operator B affiliated with N and commuting with the range of J , as well as a partial isometry w ∈ N such that w ∗ w = s ( B ) = J (1) and T ( x ) = wBJ ( x ) , ( x ∈ M ∩ L p ( M )) . Such a factorization (which is necessarily unique) is called a Yeadon type factorizationin [4, 5]. We further say that T admits a direct Yeadon type factorization if the Jordanhomomorphism J in this factorization is a ∗ -homomorphism. It is proved in [5, Proposition4.4] and [1, Theorem 3.6] that any separating map T : L p ( M ) → L p ( N ) with a directYeadon type factorization is necessarily completely bounded. It is also proved in [5,Proposition 4.5] that any such map is S -bounded (see Section 2 below for the definition).The main purpose of the present paper is to establish a form of converse of these resultsfor surjective maps. More precisely, we prove the following characterizations. Theorem.
Let 1 ≤ p < ∞ , let M , N be semifinite von Neumann algebras and let T : L p ( M ) → L p ( N ) be a surjective separating map. The following are equivalent :(i) T is S -bounded; Key words and phrases. von Neumann algebras, noncommutative L p -spaces, separating maps, operatorspaces. (ii) There exists a direct sum decomposition M = M ∞ ⊕ M such that the restrictionof T to L p ( M ) has a direct Yeadon type factorization and M is subhomogeneous.Moreover if p = 2, then (ii) is also equivalent to :(iii) T is completely bounded.These results will be proved in Section 4. We also provide an example showing that the sur-jectivity assumption cannot be dropped. In section 3, we establish a general decompositionresult for bijective separating maps which plays a key role in the above characterizationresults. We prove in passing that the inverse of any bijective separating map is separatingas well. Section 2 is preparatory. 2. Background
In this section we recall some necessary background on semifinite noncommutative L p -spaces and subhomogeneous von Neumann algebras.Let M be a semifinite von Neumann algebra with a normal semifinite faithful trace τ M . Assume that M ⊂ B ( H ) acts on some Hilbert space H . Let L ( M ) denote the ∗ -algebra of all closed densely defined (possibly unbounded) operators on H , which are τ M -measurable. Then for any 1 ≤ p < ∞ , the noncommutative L p -space associated with M can be defined as L p ( M ) := (cid:8) x ∈ L ( M ) : τ M ( | x | p ) < ∞ (cid:9) . We set k x k p := τ M ( | x | p ) p for any x ∈ L p ( M ). Then L p ( M ) equipped with k · k p is aBanach space. The reader may consult [3, 8, 12] and the references therein for details andfurther properties.We let S p , 1 ≤ p < ∞ , denote the noncommutative L p -space built upon B ( ℓ ) with itsusual trace; this is in fact the Schatten p -class of operators on ℓ . For any m ≥
1, we let S pm denote the Schatten p -class of m × m matrices. Whenever E is an operator space, welet S pm [ E ] denote the E -valued Schatten space introduced in [6, Chapter1].Recall that we may identify L p ( M ⊗ M m ) with L p ( M ) ⊗ S pm in a natural way. Let N be, possibly, another semifinite von Neumann algebra. We say that an operator T : L p ( M ) → L p ( N ) is completely bounded if there exists a constant K ≥ k T ⊗ I S pm : L p ( M ⊗ M m ) → L p ( N ⊗ M m ) k ≤ K, for any m ≥
1. In this case, the completely bounded norm of T is the smallest suchuniform bound and is denoted by k T k cb . We further say that T is a complete isometry if T ⊗ I S pm is an isometry for any m ≥ S -valued noncommutative L p -spaces, which naturallyextend previous constructions from [2, 6]. We recall this definition here.For 1 ≤ p < ∞ , the S -valued noncommutative L p -space, L p ( M ; S ), is the space of allinfinite matrices [ x ij ] i,j ≥ in L p ( M ) for which there exist families ( a ik ) i,k ≥ and ( b kj ) k,j ≥ in L p ( M ) such that P i,k a ik a ∗ ik and P k,j b ∗ kj b kj converge in L p ( M ) and for all i, j ≥ x ij = ∞ X k =1 a ik b kj . URJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES 3 We equip L p ( M ; S ) with the following norm k [ x ij ] k L p ( M ; S ) = inf (cid:13)(cid:13)(cid:13) ∞ X i,k =1 a ik a ∗ ik (cid:13)(cid:13)(cid:13) p (cid:13)(cid:13)(cid:13) ∞ X k,j =1 b ∗ kj b kj (cid:13)(cid:13)(cid:13) p , (1)where the infimum is taken over all families ( a ik ) i,k ≥ and ( b kj ) k,j ≥ as above. The space L p ( M ; S ) endowed with this norm is a Banach space.For any integer m ≥
1, we let L p ( M ; S m ) be the subspace of L p ( M ; S ) of matrices[ x ij ] i,j ≥ with support in { , . . . , m } .Following [5, Definition 3.8], we say that a bounded operator T : L p ( M ) → L p ( N ) is S -bounded if there exists a constant K ≥ k T ⊗ I S m : L p ( M ; S m ) −→ L p ( N ; S m ) k ≤ K, for any m ≥
1. In this case, the S -bounded norm of T is the smallest such uniformbounded and is denoted by k T k S . We further say that T : L p ( M ) → L p ( N ) is an S -isometry if for each m ≥ T ⊗ I S m is an isometry.We proved in [5] that for any n ≥ L p ( M n ; S m ) = S pn [ S m ] isometrically. Further, if M , N are hyperfinite, then T : L p ( M ) → L p ( N ) is S -bounded if and only if it is regularin the sense of [7].We note that any direct sum M = M ∞ ⊕ M induces isometric identifications L p ( M ) = L p ( M ) p ⊕ L p ( M ) and L p ( M ; S ) = L p ( M ; S ) p ⊕ L p ( M ; S ) (see [5, Lemma 5.2] for thelast identification).Recall that a C ∗ -algebra A is called subhomogeneous of degree ≤ N if all irreduciblerepresentations of A are of maximum dimension N . If A is subhomogeneous of degree ≤ N , for some N , we simply say that A is subhomogeneous. It is well-known (see forexample [9, Theorem 7.1.1]) that M is a subhomogeneous von Neumann algebra of degree ≤ N if and only if there exist r ≥
1, integers 1 ≤ n ≤ n ≤ . . . ≤ n r ≤ N and abelianvon Neumann algebras L ∞ (Ω ) , . . . , L ∞ (Ω r ) such that M ≃ ∞ ⊕ ≤ j ≤ r L ∞ (Ω j ; M n j ) . (2)If a von Neumann algebra M is not subhomogeneous of degree ≤ N , it is well-knownthat there is a non zero ∗ -homomorphism γ : M N +1 → M . Lemma 2.1 below makes thismore explicit in the semifinite case. Lemma 2.1.
Let M be a semifinite von Neumann algebra and let N ≥ . If M is notsubhomogeneous of degree ≤ N , then there is a complete isometry from S pN +1 into L p ( M ) that is also an S -isometry.Proof. Let M = M ∞ ⊕ M be the direct sum decomposition of M into a type I summand M and a type II summand M (see e.g. [11, Section 5]).Assume that M = { } . Following the same lines as in [5, Lemma 2.3], there is aprojection e in M , a trace preserving von Neumann algebra identification M ≃ M N +1 ⊗ ( e M e )(3) C. LE MERDY AND S. ZADEH and a finite trace projection ε in e M e such that the mapping γ : M N +1 → M ⊂ M ; γ ( a ) = a ⊗ ε is a non zero ∗ -homomorphism taking values in L ( M ), and therefore L p ( M ).For every [ a ij ] ≤ i,j ≤ m in S pN +1 ⊗ S pm we have that k [ a ij ⊗ ε ] k L p ( M ⊗ M m ) = k ε k p k [ a ij ] k L p ( M N +1 ⊗ M m ) , and therefore k ε k − p γ is a complete isometry from S pN +1 into L p ( M ). By [5, Lemma 5.1], k [ a ij ⊗ ε ] k L p ( M ; S m ) = k ε k p k [ a ij ] k S pN +1 [ S m ] , and therefore k ε k − p γ is also an S -isometry from S pN +1 into L p ( M ).If M = { } , then M is of type I. Since M is not subhomogeneous of degree ≤ N , itfollows from [11, Theorem V.1.27] that there exist a Hilbert space H with dim( H ) ≥ N + 1and an abelian von Neumann algebra W such that M contains B ( H ) ⊗ W as a summand.Using this summand instead of (3) and arguing as above we obtain the result in this caseas well. (cid:3) bijective separating maps and their inverses The goal of this section is to provide a decomposition for bijective separating maps thatfacilitates their study. We apply this decomposition to show that the inverse of a bijectiveseparating map is separating as well.First we recall some terminologies and results that we will use. A Jordan homomorphismbetween von Neumann algebras M and N is a linear map J : M → N such that J ( x ∗ ) = J ( x ) ∗ and J ( xy + yx ) = J ( x ) J ( y ) + J ( y ) J ( x )for all x and y in M . It is plain that ∗ -homomorphisms and anti- ∗ -homomorphisms are Jor-dan homomorphisms. In fact, every Jordan homomorphism is a sum of a ∗ -homomorphismand an anti- ∗ -homomorphism, as we recall here.Let J : M → N be a Jordan homomorphism and let
D ⊂ N be the w ∗ -closed C ∗ -algebra generated by J ( M ). Then J (1) is the unit of D . By e.g. [10, Theorem 3.3],there exist projections e and f in the center of D such that e + f = J (1), x J ( x ) e is a ∗ -homomorphism, and x J ( x ) f is an anti- ∗ -homomorphism. Let N = e N e and N = f N f . Define π : M → N and σ : M → N by π ( x ) = J ( x ) e and σ ( x ) = J ( x ) f ,for all x ∈ M . Then J is valued in N ∞ ⊕ N and J ( x ) = π ( x ) + σ ( x ), for all x ∈ M .Assume that M and N are semifinite von Neumann algebras and let 1 ≤ p < ∞ . In[4], inspired by Yeadon’s fundamental description of isometries between noncommutative L p -spaces, we say that a bounded operator T : L p ( M ) → L p ( N ) has a Yeadon typefactorization if there exist a w ∗ -continuous Jordan homomorphism J : M → N , a partialisometry w ∈ N , and a positive operator B affiliated with N , which satisfy the followingconditions:(a) w ∗ w = J (1) = s ( B ), the support projection of B ;(b) every spectral projection of B commutes with J ( x ), for all x ∈ M ;(c) T ( x ) = wBJ ( x ) for all x ∈ M ∩ L p ( M ). URJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES 5 We call ( w, B, J ) the Yeadon triple associated with T . This triple is unique. Following [5],if J is a ∗ -homomorphism (respectively, anti- ∗ -homomorphism), we say that T has a direct(respectively, anti-direct) Yeadon type factorization.Following [4], we say that a bounded operator T : L p ( M ) → L p ( N ) is separating if forevery x, y ∈ L p ( M ) such that x ∗ y = xy ∗ = 0, we have that T ( x ) ∗ T ( y ) = T ( x ) T ( y ) ∗ = 0.The following characterization has a fundamental role in the study of separating maps. Theorem 3.1. ([1, Theorem 3.3], [4, Theorem 3.5]) A bounded operator T : L p ( M ) → L p ( N ) admits a Yeadon type factorization if and only if it is separating. It is easy to see that for a separating map T : L p ( M ) → L p ( N ) with Yeadon triple( w, B, J ), we have that T ( z ∗ ) = wT ( z ) ∗ w ( z ∈ L p ( M )) . (4)Also, if T has a direct (respectively, anti-direct) Yeadon type factorization, we get that T ( zm ) = T ( z ) J ( m ) (respectively, T ( mz ) = T ( z ) J ( m )) , (5)for every z ∈ L p ( M ) and m ∈ M . Remark 3.2.
Let T : L p ( M ) → L p ( N ) be a separating map with Yeadon triple ( w, B, J ).We observe that if T is surjective, then w is a unitary. Indeed on the one hand, we seethat T is valued in wL p ( N ). Since ww ∗ w = w , this implies that T is valued in ww ∗ L p ( N ).Hence, if T is surjective, we have ww ∗ L p ( N ) = L p ( N ), which implies that ww ∗ = 1. Onthe other hand, T ( x ) = T ( x ) J (1), for any x ∈ L p ( M ). Hence, T is valued in L p ( N ) J (1).Hence, if T is surjective, we have L p ( N ) J (1) = L p ( N ), which implies w ∗ w = J (1) = 1. Proposition 3.3.
Let T : L p ( M ) → L p ( N ) be a separating map that is bijective. Thenthere exist direct sum decompositions M = M ∞ ⊕ M , and N = N ∞ ⊕ N , and bounded bijective separating maps T : L p ( M ) → L p ( N ) with a direct Yeadon typefactorization and T : L p ( M ) → L p ( N ) with an anti-direct Yeadon type factorizationsuch that T = T + T .Proof. Assume that w = 1. Consider a decomposition for J , induced by central projections e and f , as recalled above. As detailed in [5, Remark 4.3], this induces a decomposition N = N ∞ ⊕ N and separating maps T : L p ( M ) −→ L p ( N ) , T ( x ) = T ( x ) e, with Yeadon triple ( e, Be, π ) , and hence a direct Yeadon type factorization, and T : L p ( M ) −→ L p ( N ) , T ( x ) = T ( x ) f, with Yeadon triple ( f, Bf, σ ), and hence an anti-direct Yeadon type factorization, suchthat T = T + T .Let M := ker( σ ) and M := ker( π ). Since M and M are w ∗ -closed ideals of M ,there exist central projections α, β ∈ M such that M = α M , and M = β M . Set M := (1 − α )(1 − β ) M . Note that αβ ∈ ker( σ ) ∩ ker( π ), and therefore J ( αβ ) = 0. Since T is one-to-one, by [4, Remark 3.14(a)], J is one-to-one and therefore we must have that αβ = 0. Hence, 1 = α + β + (1 − α )(1 − β ) . C. LE MERDY AND S. ZADEH
Consequently, M = M ∞ ⊕ M ∞ ⊕ M , and so we have the decomposition L p ( M ) = L p ( M ) p ⊕ L p ( M ) p ⊕ L p ( M ) . The result will follow if we can show that L p ( M ) = ker( T ) , L p ( M ) = ker( T ) and M = { } . To see that L p ( M ) ⊆ ker( T ), let x ∈ M ∩ L p ( M ), then T ( x ) = Bσ ( x ) = 0 . Hence, M ∩ L p ( M ) ⊂ ker( T ) and therefore L p ( M ) ⊂ ker( T ). Now suppose that x belongs to ker( T ). For any n ≥
1, let p n = χ [ − n,n ] ( | x ∗ | ), the projection associated withthe indicator function of [ − n, n ] in the Borel functional calculus of | x ∗ | , and x n := p n x .Then, using (5), we have that T ( x n ) = T ( x ) σ ( p n ) = 0 . Hence, Bσ ( x n ) = 0. Since s ( B ) = 1, this implies that σ ( x n ) = 0, that is x n is in M .Now because x n → x in L p ( M ), we obtain that x belongs to L p ( M ). Hence, L p ( M ) = ker( T ) . Similarly, we can show that L p ( M ) = ker( T ).Finally, we show that M = { } . Let x ∈ L p ( M ). By surjectivity of T , there is y in L p ( M ) such that T ( y ) = T ( x ). Writing T ( y ) = T ( y ) + T ( y ), we obtain that T ( x − y ) = 0 and T ( y ) = 0, that is x − y belongs to ker( T ) = L p ( M ) and y belongsto ker( T ) = L p ( M ), thus x belongs to L p ( M ) p ⊕ L p ( M ). Hence, M = { } . Thiscompletes the proof in the case w = 1.In the general case, consider the map e T := w ∗ T ( · ), which takes any x ∈ M ∩ L p ( M ) to BJ ( x ). By Remark 3.2, e T is also a bijective separating map. Its Yeadon triple is (1 , B, J ).We may apply the above decomposition to the map e T to obtain decompositions M = M ∞ ⊕ M , N = N ∞ ⊕ N and bounded bijective separating maps f T : L p ( M ) → L p ( N )with a direct Yeadon type factorization and f T : L p ( M ) → L p ( N ) with an anti-directYeadon type factorization such that e T = f T + f T . Since w e T = T , we obtain the result. (cid:3) Proposition 3.4.
Suppose that T : L p ( M ) → L p ( N ) is a bijective separating map, then (i) T − : L p ( N ) → L p ( M ) is separating. (ii) If J : M → N is the Jordan homomorphism associated with T , then J is invertibleand J − : N → M is the Jordan homomorphism associated with T − .Proof. Using the decomposition given in Proposition 3.3, it is enough to show parts ( i ) and( ii ) for a bijective separating map with a direct Yeadon type factorization. So, throughoutthe proof we assume that this is the case. Note that by Remark 3.2, J (1) = 1.( i ) Suppose that a, b ∈ L p ( N ) such that a ∗ b = ab ∗ = 0. We show that T − ( a ) ∗ T − ( b ) = T − ( a ) T − ( b ) ∗ = 0. Let x = T − ( a ) and y = T − ( b ). Set p n := χ [ − n,n ] ( | y | ), for any URJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES 7 n ≥
1. We have that T ( x ∗ yp n ) B = T ( x ∗ ) J ( yp n ) B by (5)= T ( x ∗ ) w ∗ T ( yp n )= wT ( x ) ∗ T ( yp n ) by (4)= wT ( x ) ∗ T ( y ) J ( p n ) by (5)= wa ∗ bJ ( p n ) = 0 . Since s ( B ) = J (1) = 1, we obtain T ( x ∗ yp n ) = 0. Because T is one-to-one, we have that x ∗ yp n = 0. Now, since yp n → y , we get that x ∗ y = 0. A similar argument using ab ∗ = 0implies that xy ∗ = 0. Hence T − must be separating.( ii ) By part ( i ), T − is separating. We let J ′ denote the Jordan homomorphism of itsYeadon triple. Let e ∈ N be a projection with finite trace. For any y ∈ e N e , we havethat T − ( y ) = T − ( e ) J ′ ( y ). Applying (5), we deduce that y = T T − ( y ) = T (cid:0) T − ( e ) J ′ ( y ) (cid:1) = T T − ( e ) J J ′ ( y ) = e J J ′ ( y ) . Using the w ∗ -continuity of J and J ′ , and the w ∗ -density of the union of the e N e , for τ N ( e ) < ∞ , we deduce that y = J J ′ ( y ) for any y ∈ N . By [4, Remark 3.14(a)], since T isone-to-one, J must be one-to-one. Hence, J is invertible with J − = J ′ . (cid:3) Remark 3.5.
Part ( ii ) of Proposition 3.4 shows that a separating invertible map T : L p ( M ) → L p ( N ) admits a direct Yeadon type factorization if and only if T − does.4. a characterization of completely/ S -bounded surjective separatingmaps In this section we show that a separating map can always be reduced to a one-to-oneseparating map and therefore we may confine ourself to the study of separating maps thatare surjective rather than bijective. The goal of the section is to provide a characterizationfor surjective separating maps that are completely bounded (when p = 2) or S -bounded.We show that the surjectivity assumption is essential.We require [5, Propositions 4.4 & 4.5] later on in our arguments in this section. Werecall the statements for convenience. Proposition 4.1.
Let T : L p ( M ) → L p ( N ) be a bounded operator with a direct Yeadontype factorization. Then T is completely bounded and k T k cb = k T k . Proposition 4.2.
Let T : L p ( M ) → L p ( N ) be a bounded operator with a direct Yeadontype factorization. Then T is S -bounded and k T k S = k T k . Lemma 4.3.
Let T : L p ( M ) → L p ( N ) be a separating map. Then there exists a directsum decomposition M = M ∞ ⊕ f M such that ker ( T ) = L p ( M ) .Proof. Let T : L p ( M ) → L p ( N ) be a separating map and J : M → N be the Jordanhomomorphism associated with T via its Yeadon type factorization. Let M := ker( J ).Then M is an ideal. Since J is w ∗ -continuous, M is w ∗ -closed. Hence we have a directsum decomposition M = M ∞ ⊕ f M . C. LE MERDY AND S. ZADEH
It is clear that L p ( M ) ⊂ ker T . Further J | f M is one-to-one. By [4, Remark 3.14(a)] thisimplies that T | L p ( f M ) is one-to-one. This yields the result. (cid:3) For any von Neumann algebra M , we let M op denote its opposite von Neumann algebra.Recall that the underlying dual Banach space structure and involution on M op are thesame as on M but the product of x and y is defined by yx rather than xy . Note that theBanach spaces L p ( M ) and L p ( M op ) are the same. It is evident that, for von Neumannalgebras M and N , J : M → N is a ∗ -homomorphism if and only if J op : M op → N ; x J ( x ) , is an anti- ∗ -homomorphism. Hence, a separating map T : L p ( M ) → L p ( N ) has a directYeadon type factorization if and only if T op : L p ( M op ) → L p ( N ); x T ( x ) , has an anti-direct Yeadon type factorization.Lemma 4.4 below is the principal ingredient of our characterization theorems. Its proofrelies on the relation between the completely bounded norm or S -norm of the identitymap I op : L p ( M ) → L p ( M op )and the norms of the transformations[ x ij ] ≤ i,j ≤ m [ x ji ] ≤ i,j ≤ m either on L p ( M ⊗ M m ) or on L p ( M ; S m ), in particular in the specific case when M = M n .We will use the fact that for any n ≥
1, we have L p ( M n ⊗ M m ) ≃ S pm [ S pn ], isometrically,provided that S pn is equipped with the operator space structure given in [6].Let t m denote the transposition map on scalar m × m matrices. Assume that M issemifinite. The map I M op ⊗ t m : M op ⊗ M m → M op ⊗ M opm is a trace preserving ∗ -homomorphism, and so I L p ( M op ) ⊗ t m : L p ( M op ⊗ M m ) −→ L p ( M op ⊗ M opm )is an isometry. Moreover M op ⊗ M opm = ( M⊗ M m ) op , hence L p ( M op ⊗ M opm ) = L p ( M⊗ M m )isometrically. For any [ x ij ] ≤ i,j ≤ m in L p ( M ) ⊗ S pm , since I L p ( M op ) ⊗ t m maps [ x ij ] to [ x ji ],we get that(6) (cid:13)(cid:13) [ x ij ] (cid:13)(cid:13) L p ( M op ⊗ M m ) = (cid:13)(cid:13) [ x ji ] (cid:13)(cid:13) L p ( M ⊗ M m ) . We now show that similarly, for any [ x ij ] ≤ i,j ≤ m in L p ( M ) ⊗ S m ,(7) (cid:13)(cid:13) [ x ij ] (cid:13)(cid:13) L p ( M op ; S m ) = (cid:13)(cid:13) [ x ji ] (cid:13)(cid:13) L p ( M ; S m ) . To verify the identity (7), assume that k [ x ij ] k L p ( M op ; S m ) <
1. Taking into account theopposite product and (1), we can write x ij = X k b kj a ik for some a ik , b kj in L p ( M ) such that P i,k a ∗ ik a ik and P k,j b kj b ∗ kj have norm < L p ( M ).This exacly means that k [ x ji ] k L p ( M ; S m ) <
1. This shows the inequality ≥ in (7). Reversingthe argument we find the other inequality. URJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES 9 Identities (6) and (7), respectively, imply(8) (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) cb = sup m ≥ (cid:13)(cid:13) I L p ( M ) ⊗ t m : L p ( M⊗ M m ) −→ L p ( M⊗ M m ) (cid:13)(cid:13) , and(9) (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) S = sup m ≥ (cid:13)(cid:13) I L p ( M ) ⊗ t m : L p ( M ; S m ) −→ L p ( M ; S m ) (cid:13)(cid:13) . When M = M n , the above identities can be more specific. In fact, as we show below,we have that for any n ≥ (cid:13)(cid:13) I op : S pn −→ { S pn } op (cid:13)(cid:13) cb = (cid:13)(cid:13) t n : S pn → S pn (cid:13)(cid:13) cb (10)and (cid:13)(cid:13) I op : S pn −→ { S pn } op (cid:13)(cid:13) S = (cid:13)(cid:13) t n : S pn → S pn (cid:13)(cid:13) S . (11)Using (6) applied to M = M n , to prove (10), it is enough to show that for any [ x ij ] ≤ i,j ≤ m in S pn ⊗ S pm ,(12) (cid:13)(cid:13) [ t n ( x ij )] (cid:13)(cid:13) S pm [ S pn ] = (cid:13)(cid:13) [ x ji ] (cid:13)(cid:13) S pm [ S pn ] . This follows from the fact that t m ⊗ t n = t nm is an isometry on S pm [ S pn ] ≃ S pnm , and hence (cid:13)(cid:13) ( t m ⊗ t n )[ t n ( x ij )] (cid:13)(cid:13) S pm [ S pn ] = (cid:13)(cid:13) [ t n ( x ij )] (cid:13)(cid:13) S pm [ S pn ] . Since ( t m ⊗ t n )[ t n ( x ij )] = [ x ji ], this yields (12).Likewise, using (7) applied to M = M n , to prove (11), it is enough to show that forany [ x ij ] ≤ i,j ≤ m in S pn ⊗ S m ,(13) (cid:13)(cid:13) [ t n ( x ij )] (cid:13)(cid:13) S pn [ S m ] = (cid:13)(cid:13) [ x ji ] (cid:13)(cid:13) S pn [ S m ] . Assume that k [ t n ( x ij )] k S pn [ S m ] <
1. According to (1), we can write t n ( x ij ) = X k a ik b kj for some a ik , b kj in S pn such that P i,k a ik a ∗ ik and P k,j b ∗ kj b kj have norm < S pn . Thenwe have x ij = X k t n ( a ik b kj ) = X k t n ( b kj ) t n ( a ik ) , hence x ji = X k t n ( b ki ) t n ( a jk ) . Further X k,j t n ( a jk ) ∗ t n ( a jk ) = t n (cid:16)X j,k a jk a ∗ jk (cid:17) , and t n is an isomerty on S pn . Consequently, P k,j t n ( a kj ) ∗ t n ( a jk ) has norm < S pn .Similarly, P i,k t n ( b ki ) t n ( b ki ) ∗ has norm < S pn . This shows that k [ x ji ] k S pn [ S m ] < ≥ in (13). Reversing the argument we find the otherinequality.In the sequel, E ( x ) denotes the integer part of x . Lemma 4.4.
Suppose that M is a semifinite von Neumann algebra. ( i ) If M is subhomogeneous of degree ≤ N for some N ≥ , then for all [ x ij ] ∈ M m ⊗ L p ( M ) , m ≥ , we have that k [ x ji ] k L p ( M⊗ M m ) ≤ N | / − /p | k [ x ij ] k L p ( M⊗ M m ) , and k [ x ji ] k L p ( M ; S m ) ≤ N k [ x ij ] k L p ( M ; S m ) . ( ii ) Suppose that there exists K ≥ such that for all [ x ij ] ∈ L p ( M ) ⊗ S pm , m ≥ , k [ x ji ] k L p ( M⊗ M m ) ≤ K k [ x ij ] k L p ( M⊗ M m ) . (14) Then if p = 2 , M is subhomogeneous of degree ≤ N with N = E (cid:16) K | / − /p | (cid:17) . ( iii ) Suppose that there exists K ≥ such that for all [ x ij ] ∈ L p ( M ) ⊗ S pm , m ≥ , k [ x ji ] k L p ( M ; S m ) ≤ K k [ x ij ] k L p ( M ; S m ) . (15) Then M is subhomogeneous of degree ≤ N with N = E ( K ) .Proof. ( i ) Assume that M = L ∞ (Ω; M n ). Let m ≥ L p ( M ⊗ M m ) ≃ L p (Ω; S pm [ S pn ]) . By Pisier-Fubini Theorem [6, (3.6)], L p ( M ; S m ) ≃ L p (Ω; S pn [ S m ]) . Consequently, (cid:13)(cid:13) I L p ( M ) ⊗ t m : L p ( M ⊗ M m ) −→ L p ( M ⊗ M m ) (cid:13)(cid:13) = (cid:13)(cid:13) t m ⊗ I S pn : S pm [ S pn ] −→ S pm [ S pn ] (cid:13)(cid:13) . (16)and (cid:13)(cid:13) I L p ( M ) ⊗ t m : L p ( M ; S m ) −→ L p ( M ; S m ) (cid:13)(cid:13) = (cid:13)(cid:13) I S pn ⊗ t m : S pn [ S m ] −→ S pn [ S m ] (cid:13)(cid:13) . (17)Applying (8) to both sides of (16), we deduce (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) cb = (cid:13)(cid:13) I op : S pn −→ { S pn } op (cid:13)(cid:13) cb , and applying (9) to both sides of (17), we deduce that (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) S = (cid:13)(cid:13) I op : S pn −→ { S pn } op (cid:13)(cid:13) S . By [5, Lemma 5.3], (cid:13)(cid:13) t n : S pn → S pn (cid:13)(cid:13) cb = n | /p − / | and (cid:13)(cid:13) t n : S pn → S pn (cid:13)(cid:13) S = n, hence we obtain by (10) and (11) that (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) cb = n | /p − / | and (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) S = n. When M is subhomogeneous of degree ≤ N , there exist r ≥
1, integers 1 ≤ n ≤ n ≤ · · · ≤ n r ≤ N and abelian von Neumann algebras L ∞ (Ω ) , . . . , L ∞ (Ω r ) such that (2)holds. Then for any m ≥
1, we have that L p ( M ⊗ M m ) ≃ p ⊕ ≤ j ≤ r L p (Ω j ; S pm [ S pn j ]) and L p ( M ; S m ) ≃ p ⊕ ≤ j ≤ r L p (Ω j ; S pn j [ S m ]) . URJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES 11 Using our previous argument and direct sums we deduce that k I op : L p ( M ) → L p ( M op ) k cb ≤ N | p − | and k I op : L p ( M ) → L p ( M op ) k S ≤ N. The result follows from (6) and (7).( ii ) Suppose that M is not subhomogeneous of degree ≤ N = E ( K | / − /p | ). ByLemma 2.1, there exists a complete isometry S pN +1 ֒ → M . This embedding implies that for any m ≥ (cid:13)(cid:13) t m ⊗ I S pN +1 : S pm [ S pN +1 ] −→ S pm [ S pN +1 ] (cid:13)(cid:13) ≤ (cid:13)(cid:13) I L p ( M ) ⊗ t m : L p ( M⊗ M m ) −→ L p ( M⊗ M m ) (cid:13)(cid:13) . According to (8) and (10), this implies that (cid:13)(cid:13) t N +1 : S pN +1 −→ S pN +1 (cid:13)(cid:13) cb ≤ (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) cb . Hence (cid:13)(cid:13) I op : L p ( M ) −→ L p ( M op ) (cid:13)(cid:13) cb ≥ ( N + 1) | p − | . Comparing this with inequality (14) above and applying (6), we get a contradiction.( iii ) The proof is similar to the proof of part ( ii ). (cid:3) Proposition 4.5.
Let T : L p ( M ) → L p ( N ) be separating. If M is subhomogeneous then T is completely bounded and S -bounded.Proof. Changing T to w ∗ T , we can assume that w = J (1). By [5, Remark 4.3], wecan write T as a sum T = T + T such that T has a direct Yeadon type factorizationand T has an anti-direct Yeadon type factorization. By Propositions 4.1 and 4.2, T is completely bounded and S -bounded. Hence it suffices to show that T is completelybounded and S -bounded. Let I op : L p ( M ) → L p ( M op ) be the identity map and set T op =: T ◦ I op − . Since T has an anti-direct Yeadon type factorization, T op has a directYeadon type factorization. So, by Propositions 4.1 and 4.2, T op is completely bounded and S -bounded. Since M is subhomogeneous, part ( i ) of Lemma 4.4 and its proof show that I op is completely bounded and S -bounded. By composition, we obtain that T = T op ◦ I op is completely bounded and S -bounded. (cid:3) Proposition 4.6.
Suppose that T : L p ( M ) → L p ( N ) is a bijective separating map withan anti-direct Yeadon type factorization. (i) If p = 2 and T is completely bounded then M is subhomogeneous. (ii) If T is S -bounded then M is subhomogeneous.Proof. ( i ) Suppose that T : L p ( M ) → L p ( N ), 1 ≤ p = 2 < ∞ , is a bijective separatingmap with an anti-direct Yeadon type factorization. Assume that T is completely bounded.Let I op : L p ( M ) → L p ( M op ) be the identity map and set T op := T ◦ I op − . Since T is bijective with an anti-direct Yeadon type factorization, T op is bijective with a directYeadon type factorization. By part ( i ) of Proposition 3.4 and Remark 3.5, T op − is alsoseparating with a direct Yeadon type factorization. Therefore, by Proposition 4.1, T op − is completely bounded. Hence, I op := T op − ◦ T is completely bounded. It now followsfrom part ( ii ) of Lemma 4.4 and (6) that M is subhomogeneous.( ii ) The same argument as in part ( i ) with S -bounded (norm) replacing completelybounded (norm), Proposition 4.2 replacing Proposition 4.1, part ( iii ) of Lemma 4.4 re-placing its part ( ii ) and (7) replacing (6) yields the result. (cid:3) Remark 4.7.
Suppose that T : L p ( M ) → L p ( N ), 1 ≤ p < ∞ , is a surjective separatingisometry with an anti-direct Yeadon type factorization. The proof of Proposition 4.6shows that when T is completely bounded and p = 2, M is subhomogeneous of degree ≤ E ( k T k | / − /p | cb ). When T is S -bounded, M is subhomogeneous of degree ≤ E ( k T k S ). Theorem 4.8.
Let T : L p ( M ) → L p ( N ) , ≤ p = 2 < ∞ , be a bounded separating mapthat is surjective. Then the following are equivalent. (i) T is completely bounded. (ii) There exists a decomposition M = M ∞ ⊕ M such that T | L p ( M ) has a directYeadon type factorization and M is subhomogeneous.Proof. ( i ) = ⇒ ( ii ) Suppose that T : L p ( M ) → L p ( N ), 1 ≤ p = 2 < ∞ , is a surjectivecompletely bounded separating map. In view of Lemma 4.3, we may assume T is bijective.By Proposition 3.3, there exist decompositions M = M ∞ ⊕ M and N = N ∞ ⊕ N andsurjective separating maps T : L p ( M ) → L p ( N ) and T : L p ( M ) → L p ( N ) such that T has a direct Yeadon type factorization, T has an anti-direct Yeadon type factorizationand T = T + T . Since T is completely bounded, T is also completely bounded. By part( i ) of Proposition 4.6, M must be subhomogeneous.( ii ) = ⇒ ( i ) This is a consequence of Propositions 4.1 and 4.5. (cid:3) Theorem 4.9.
Let T : L p ( M ) → L p ( N ) , ≤ p < ∞ , be a separating map that issurjective. Then the following are equivalent. (i) T is S -bounded. (ii) There exists a decomposition M = M ∞ ⊕ M such that T | L p ( M ) has a directYeadon type factorization and M is subhomogeneous.Proof. The proof is similar to Theorem 4.8, replacing completely bounded with S -bounded,part ( i ) of Proposition 4.6 by its part ( ii ) and Proposition 4.1 by Proposition 4.2. (cid:3) The following example shows the surjectivity assumption in Theorems 4.8 and 4.9 isessential. In fact in this example, on a non-subhomogeneous semifinite von Neumannalgebra M and for a given ε >
0, we construct a separating isometry T : L p ( M ) → L p ( N )such that T is not surjective, k T k cb ≤ ε , k T k S ≤ ε and part ( ii ) of Theorems 4.8and 4.9 is not satisfied.The isometry T in our example is set up between hyperfinite von Neumann algebrasand so k T k cb ≤ k T k S (see [7, Proposition 2.2] and [5, Proposition 3.11]). Therefore, weonly need to verify that for such T we have that k T k S ≤ ε . Example 4.10.
Let 1 < p < ∞ . Consider the von Neumman algebra M = ℓ ∞ { M n } = (cid:8) ( x n ) n ≥ : ∀ n ≥ , x n ∈ M n and sup n ≥ k x n k ∞ < ∞ (cid:9) , the infinite direct sum of all M n , n ≥
1. Let N := M ∞ ⊕ M , the direct sum of two copiesof M . The noncommutative L p -space associated with M is ℓ p { S pn } = (cid:8) ( x n ) n ≥ : ∀ n ≥ , x n ∈ S pn and X n ≥ k x n k pp < ∞ (cid:9) , URJECTIVE SEPARATING MAPS ON NONCOMMUTATIVE L p -SPACES 13 equipped with the norm (cid:13)(cid:13) ( x n ) n ≥ (cid:13)(cid:13) p = (cid:16) ∞ X n =1 k x n k pp (cid:17) p , and so the noncommutative L p -space associated with N is ℓ p { S pn } p ⊕ ℓ p { S pn } . Let ( β n ) n ≥ be a sequence in the interval (0 , T : ℓ p { S pn } → ℓ p { S pn } and T : ℓ p { S pn } → ℓ p { S pn } by setting T (cid:0) ( x n ) n ≥ (cid:1) = (cid:0) (1 − β n ) p x n (cid:1) n ≥ and T (cid:0) ( x n ) n ≥ (cid:1) = (cid:0) β p n t n ( x n ) (cid:1) n ≥ for any x = ( x n ) n ≥ ∈ ℓ p { S pn } . Consider T : ℓ p { S pn } → ℓ p { S pn } p ⊕ ℓ p { S pn } , T ( x ) = ( T ( x ) , T ( x )) . It is plain that T is an isometry. Indeed for any x = ( x n ) n ≥ ∈ ℓ p { S pn } , we have k T ( x ) k pp = k T ( x ) k pp + k T ( x ) k pp = ∞ X n =1 (1 − β n ) k x n k pp + ∞ X n =1 β n k t x n k pp = ∞ X n =1 k x n k pp = k x k pp . Given ε >
0, consider the above construction with β n = (1 + ε ) p − n p − . We show that T is S -bounded with k T k S ≤ ε . Indeed consider an integer m ≥ ℓ p { S pn } (cid:2) S m (cid:3) = ℓ p { S pn [ S m ] } , and therefore, we also have that (cid:18) ℓ p { S pn } p ⊕ ℓ p { S pn } (cid:19) (cid:2) S m (cid:3) = ℓ p { S pn [ S m ] } p ⊕ ℓ p { S pn [ S m ] } . Now let x = ( x n ) n ≥ ∈ ℓ p { S pn [ S m ] } (here each x n is an element of S pn [ S m ]). Then( I S m ⊗ T )( x ) = (cid:16)(cid:0) (1 − β n ) p x n (cid:1) n ≥ , (cid:0) β p n ( t n ⊗ I S m )( x n ) (cid:1) n ≥ (cid:17) . Consequently, (cid:13)(cid:13) ( I S m ⊗ T )( x ) (cid:13)(cid:13) pp = ∞ X n =1 (1 − β n ) k x n k pS pn [ S m ] + ∞ X n =1 β n k ( t n ⊗ I S m )( x n ) k pS pn [ S m ] ≤ ∞ X n =1 (1 − β n ) k x n k pS pn [ S m ] + n p β n k x n k pS pn [ S m ] by [5, Lemma 5.3 (ii)] ≤ (1 + ε ) p ∞ X n =1 k x n k pS pn [ S m ] = (1 + ε ) p k x k pp . It is clear that T is separating and that the Jordan homomorphism J : M → N in itsYeadon triple is given by J (cid:0) ( x n ) n ≥ (cid:1) = (cid:0) ( x n ) n ≥ , ( t n ( x n )) n ≥ (cid:1) . It follows that whenever M is a non zero summand of M , the Yeadon factorization ofthe restriction of T to L p ( M ) is neither direct nor indirect. A fortiori, T does not satisfythe assertion ( ii ) of Theorem 4.9. acknowledgment The second author would like to gratefully thank “Laboratoire de Math´ematiques deBesan¸con” for hospitality and excellent working condition she received during her visit.
References [1]
G. Hong, S. K. Ray and S. Wang , Maximal ergodic inequalities for positive operators onnoncommutative L p -spaces , Preprint 2020, arXiv:1907.12967v5.[2] M. Junge , Doob’s inequality for non-commutative martingales , J. Reine Angew. Math. 549(2002) 149-190.[3]
M. Junge and Q. Xu , Noncommutative maximal ergodic theorems , J. Amer. Math. Soc. 20(2007), 385-439.[4]
C. Le Merdy and S. Zadeh , ℓ -contractive maps on noncommutative L p -spaces , to appear inJournal of Operator Theory, arXiv:1907.03995.[5] C. Le Merdy and S. Zadeh , On factorization of separating maps on noncommutative L p -spaces , preprint 2020, arXiv:2007.04577.[6] G. Pisier , Non-commutative vector valued L p -spaces and completely p -summing maps ,Ast´erisque, Vol. 247, 1998.[7] G. Pisier , Regular operators between non-commutative L p -spaces , Bull. Sci. Math. 119 (1995),95-118.[8] G. Pisier and Q. Xu , Non-commutative L p -spaces , Handbook of the geometry of Banachspaces, Vol. 2, pp. 1459-1517, North-Holland, Amsterdam, 2003.[9] V. Runde , Amenable Banach algebras , Springer-Verlag, New York, 2020.[10]
E. Størmer , On the Jordan structure of C ∗ -algebras , Trans. Amer. Math. Soc. 120 (1965),438-447.[11] M. Takesaki , Theory of operator algebras. I , Encyclopaedia of Mathematical Sciences, Vol.124, Springer-Verlag, Berlin, 2002.[12]
M. Terp , L p -spaces associated with von Neumann algebras , Notes, Math. Institute, CopenhagenUniv., 1981.[13] F. Y. Yeadon , Isometries of noncommutative L p -spaces , Math. Proc. Cambridge Philos. Soc.90 (1981), 41-50. Laboratoire de Math´ematiques de Besanc¸on, Universite Bourgogne Franche-Comt´e, France
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