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2 0 Normal contractive projections preserve type 0 2 n a Cho-Ho Chu∗, Matthew Neal† and Bernard Russo† J 1 2 ] Abstract A O Given a JBW*-triple Z and a normal contractive projection P : h. Z −→ Z, we show that the (Murray-von Neumann) type of each at summand of P(Z) is dominated by the type of Z. m [ Introduction 1 v Contractive projections play a useful role in the theory of operator al- 8 gebras and Banach spaces. The ranges of contractive projections on C*- 8 1 algebras form an important subclass of those complex Banach spaces whose 1 open unit balls are bounded symmetric domains. An important feature of 0 2 these spaces is that they are equipped with a Jordan triple product, in- 0 duced by the Lie algebra of the automorphism group of the open unit ball. / h Known as JB∗-triples, they have been shown to be the appropriate category t a in which to study contractive projections; indeed the fact that the category m of JB∗-triples is stable under contractive projections played a key role in : v their structure theory. i X Recently, contractive projections on von Neumann algebras have arisen r in the study of operator spaces as well as the theory of harmonic functions a on locally compact groups. In [22], a family of Hilbertian operator spaces were studied and used to classify, in an appropriate sense, the ranges of contractive projections on B(H) which are atomic as Banach spaces. In [5], it was shown that the Banach space of bounded matrix-valued harmonic functions on a locally compact group is the range of a contractive projection ∗Partially supported by Ministerio de Education y Cultura, Spain, grant PB 98-1371 †Supported in part by NSF grant DMS-0101153 2000 Mathematics Subject Classiﬁcation: 46L70, 46L10, 17C65, 32M15 1 on a type I ﬁnite von Neumann algebra. It has also been shown in [6] that the Banach space of harmonic functionals on the Fourier algebra of a locally compact group G is the range of a contractive projection on the group von Neumann algebra VN(G). There is a Murray-von Neumann type classiﬁcation for JBW∗-triples, that is, JB∗-triples which are the dual of a Banach space. In view of the fact,notedabove, thattherangeofacontractiveprojectiononaJB∗-tripleis againaJB∗-triple[12,21,27],theaboveinvestigations pointtoanaturaland important question, namely, how is the Murray-von Neumann classiﬁcation of the domain aﬀected by a contractive projection? More precisely, given a JBW∗-triple Z of type X, where X = I,II, or III, is the range of a normal contractive projection on Z of type Y with Y ≤ X, meaning each summandoftherangeisoftype≤ X ? Inthispaper, weanswer thisquestion aﬃrmatively. We shall see that it suﬃces to prove this for JW∗-triples, that is, for JBW∗-triples which are linearly isometric to a weak operator closed subspace of B(H), stable for the triple product xy∗z +zy∗x, where B(H) is the von Neumann algebra of bounded operators on a Hilbert space H. Tomiyama [31] has analysed the type structure of the range of a con- tractive projection which is a von Neumann subalgebra of the domain. His arguments depend on the crucial fact that the range is a subalgebra. In our investigation, the range, which automatically has an algebraic structure, need not be a subalgebra nor even a subtriple. This adds both generality and complexity to our question. This paper is organized as follows. Section 1 is devoted to background and motivation for the problem. In section 2 we consider, as a preliminary tool, contractive projections onJW∗-algebras. Propositions 2.5 and2.6 show that if the image of a normal contractive projection on a JW∗-algebra is a JW∗-subalgebra (not necessarily with the same identity), then the properties of being semiﬁnite or of type I are passed on from the domain to the image. In section 3 we study normal contractive projections on a von Neumann algebra of type I and show in Proposition 3.5 that the image is isometric to a JW∗-triple of type I. It is necessary ﬁrst to prove this (in Proposition 3.1) in the special case when the projection is thePeirce 2-projectionwith respect to a partial isometry. Our main results, that normal contractive projections on JW∗-triples preserve both type I and semiﬁniteness, appear in section 4 as Theorems 4.2 and 4.4. Again, Propositions 4.1 and 4.3 deal with the special case of a Peirce 2-projection. Although Propositions 2.5 through 4.3 are each 2 a special case of Theorem 4.2 or 4.4, they are essential steps in the proofs of these theorems and they are new and of interest. In section 5, we extend Theorems 4.2 and 4.4 to arbitrary JBW∗-triples, and consider the case of atomic JBW∗-triples. 1 Motivation and Background Let M bea vonNeumann algebra andlet N bea vonNeumann subalgebra of M containing the identity element of M. A positive linear map E : M → N satisfying Ex = x for x ∈ N and E(axb) = aE(x)b for x ∈ M and a,b ∈ N is called a conditional expectation. Conditional expectations have played some fundamental roles in the theory of von Neumann algebras, for instance in V. Jones’ theory of subfactors. Work in the 1950s of Tomiyama and Nakamura- Takesaki-Umegaki established that conditional expectations are idempotent, contractive, and completely positive mappings, and they preserve type when normal; see the survey paper of Stormer [29]. Conversely ([19, 10.5.85]), a unitalcontractiveprojectionfromoneC∗-algebraontoaunitalC∗-subalgebra extends to a normal conditional expectation on the universal enveloping von Neumann algebra, and is in particular a conditional expectation on the C∗- algebra. A type theory for weakly closed Jordan operator algebras, based on mod- ularity of the lattice of projections, and parallel to the type classiﬁcation the- ory for von Neumann algebras, was introduced and developed in the 1960s by Topping [32] and Stormer [28]. In particular, Stormer showed that a JW- algebra is of type I if and only if its enveloping von Neumann algebra is of type I. This was extended to types II and III by Ayupov in 1982 [1]. In some cases the JW-algebra in these results is required to be reversible. A special case of a result of Choi-Eﬀros in 1977 [3], of fundamental im- portance in the rapidly advancing theory of operator spaces, states that the range of a unital completely positive projection on a C∗-algebra, while not in general a subalgebra, nevertheless carries the structure of a C∗-algebra. The proof hinges on a conditional expectation formula (needed to prove that the abstract product is associative) which is established using the Kadison- Schwarz inequality for positive linear maps. We note such a projection is completely contractive. A special case of a result of Eﬀros-Stormer in 1979 [8] states that the 3 range of a unital positive projection on a C∗-algebra, while not in general a Jordan subalgebra, carries a natural Jordan algebra structure. As before, the proof depends on a conditional expectation formula (needed to prove that the abstract product satisﬁes the Jordan identity), and such a projection is contractive. The above results raised the question of what algebraic structure existed in the range of an arbitrary contractive projection on a C∗-algebra. A special case of a result of Friedman and Russo in 1983 states that the range of such a projection is linearly isometric to a subspace, closed under the triple product xy∗z + zy∗x, of the second dual of the C∗-algebra. Because of the lack of an order structure and hence the unavailability of the Kadison-Schwarz inequality, new techniques were needed and developed by Friedman-Russo in their theory of “operator algebras without order” ([10]), including some conditional expectation formulas for the triple product ([11, Corollary 1]). Duringthe1980s,thetheoryofJB∗-tripleswas developed extensively; for a summary, see the survey [24]. In particular, a type I theory was developed for JBW∗-triples by Horn in his thesis in 1984. In this theory, idempotents (projections) were replaced by tripotents (which are abstraction of partial isometries), and the reduced algebra pAp was replaced by the Peirce 2-space of a tripotent. Of special importance here is the algebraic fact that such a Peirce 2-space has an abstract structure of a Jordan algebra, and moreover Horn has proved that a JBW∗-triple is of type I if, and only if, it contains a complete tripotent whose Peirce 2-space is a Jordan algebra of type I. The remarkable structure theorem of Horn states that type I JBW∗-triples are isometric to direct sums of tensor products of a commutative von Neumann algebra by a Cartan factor. We now recall some deﬁnitions. A Jordan triple system is a complex vector space V with a Jordan triple product {·,·,·} : V ×V ×V −→ V which is symmetric and linear in the outer variables, conjugate linear in the middle variable and satisﬁes the Jordan triple identity {a,b,{x,y,z}} = {{a,b,x},y,z}−{x,{b,a,y},z}+{x,y,{a,b,z}}. A complex Banach space Z is called a JB∗-triple if it is a Jordan triple system such that for each z ∈ Z, the linear map z(cid:3)z : v ∈ Z 7→ {z,z,v} ∈ Z 4 is Hermitian, that is, keit(z(cid:3)z)k = 1 for all t ∈ R, with non-negative spectrum and kz(cid:3)zk = kzk2. A JB∗-triple Z is called a JBW∗-triple if it is a dual Banach space, in which case its predual is unique, denoted by Z , and the ∗ triple product is separately weak* continuous. The second dual Z∗∗ of a JB∗-triple is a JBW∗-triple. A norm-closed subspace of a JB*-triple is called a subtriple if it is closed with respect to the triple product. A JBW*-triple is called a JW*-triple if it can be embedded as a subtriple of some B(H). TheJB∗-triplesformalargeclassofBanachspaceswhichincludeC∗-algebras, Hilbert spaces and spaces of rectangular matrices. The triple product in a C*-algebra A is given by 1 {x,y,z} = (xy∗z +zy∗x). 2 In fact, A is a Jordan algebra in the product 1 x◦y = (xy +yx) 2 and we have {x,y,z} = (x ◦ y∗) ◦ z + (y∗ ◦ z) ◦ x − (z ◦ x) ◦ y∗. A norm- closed subspace of a C∗-algebra is called a JC*-algebra if it is also closed with respect to the involution ∗ and the Jordan product ◦ given above. A JC∗-algebra is called a JW∗-algebra if it is a dual Banach space. An element e in a JB*-triple Z is called a tripotent if {e,e,e} = e in which case the map e(cid:3)e : Z −→ Z has eigenvalues 0, 1 and 1, and we have 2 the following decomposition in terms of eigenspaces Z = Z (e)⊕Z (e)⊕Z (e) 2 1 0 which is called the Peirce decomposition of Z. The k-eigenspace Z (e) is 2 k called the Peirce k-space. The Peirce projections from Z onto the Peirce k-spaces are given by P (e) = Q2(e), P (e) = 2(e(cid:3)e−Q2(e)), P (e) = I −2e(cid:3)e+Q2(e) 2 1 0 where Q(e)z = {e,z,e} for z ∈ Z. The Peirce projections are contractive. In later computation, we will use frequently the Peirce rules {Z (e)Z (e)Z (e)} ⊂ Z (e) i j k i−j+k 5 where Z (e) = {0} for l 6= 0,1,2. We note that the Peirce 2-space Z (e) = l 2 P (e)(Z) is a Jordan Banach algebra with identity e, the Jordan product 2 a◦b = {a,e,b} and involution a# = {e,a,e} which satisfy ka#k = kak; k{a,a#,a}k = kak3 where {x,y,z} = (x◦y∗)◦z+(y∗◦z)◦x−(z◦x)◦y∗, in other words, Z (e) is a 2 unitalJB*-algebra. AJB*-algebrahavingapredualiscalledaJBW*-algebra. As shown in [33], the self-adjoint parts of JB*-algebras(resp JBW*-algebras) are exactly the JB-algebras (resp JBW-algebras). For deﬁnitions and basic results about JB-algebras, we refer the reader to [15]. If Z = Z (e), then e is 2 called unitary. If Z (e) = {0}, then the tripotent e is called complete. Two 0 tripotents u and v are said to be orthogonal if u(cid:3)v = 0. The elements of the predual Z of a JBW*-triple Z are exactly the normal functionals on Z, that ∗ is, the continuous linear functionals on Z which are additive on orthogonal tripotents. Given an orthogonal family of tripotents {e } in a JB*-triple Z, we i i∈Λ can form a joint Peirce decomposition Z = Z ij i,j∈Λ M where Peirce spaces Z are deﬁned by ij Z = Z (e ), Z = Z (e )∩Z (e ) (i 6= j) ii 2 i ij 1 i 1 j Z = Z (e )∩ Z (e ), Z = Z (e ). i0 1 i 0 j 00 0 i j6=i i \ \ We have, for z ∈ Z and e = e , ij ij i P 0 if k ∈/ {i,j} (e (cid:3)e)(z ) = (e (cid:3)e )(z ) = k ij k k ij 1z if k ∈ {i,j}. ( 2 ij JBW*-triples have an abundance of tripotents. In fact, given a JBW*- triple Z and f in the predual Z , there is a unique tripotent v ∈ Z, called ∗ f the support tripotentof f, such that f◦P (v ) = f and the restriction f| 2 f Z2(vf) is a faithful positive normal functional. TheMurray-vonNeumannclassiﬁcationofthevonNeumannalgebrascan be extended to that of JBW*-triples and, a JBW*-triple can be decomposed 6 into a direct sum of type j (j = I,II,III) summands (see [16, 18]). A JBW*-triple is called continuous if it does not contain a type I summand in which case, it is a direct sum of a JW*-algebra H(A,α) and a weak* closed right ideal of a continuous von Neumann algebra, as shown in [18], where H(A,α) = {a ∈ A : α(a) = a} is the ﬁxed-point set of a period 2 weak* continuous antiautomorphism α of a von Neumann algebra A. It follows that continuous JBW*-triples are JW*-triples. A JBW*-triple Z is called type I if it contains an abelian tripotent e such that Z = U(e) where U(e) denotes the weak* closed triple ideal generated by e. We recall that a tripotent e is said to be abelian if the Peirce 2-space P (e)(Z) is an abelian triple which is equivalent to saying that P (e)(Z) is 2 2 an associative JBW*-algebra in the usual Jordan product x ◦y = {x,e,y}. Horn [17, 4.14] has shown that a JBW*-triple is type I if, and only if, every weak*-closed triple ideal contains an abelian tripotent. An important class of type I JBW*-triples are the following six types of Cartan factors: type 1 B(H,K) with triple product {x,y,z} = 1(xy∗z +zy∗x), 2 type 2 {z ∈ B(H,H) : zt = −z}, type 3 {z ∈ B(H,H) : zt = z}, type 4 spin factor, type 5 M (O) with triple product {x,y,z} = 1(x(y∗z)+z(y∗x)), 1,2 2 type 6 M (O) 3 where B(H,K) is the Banach space of bounded linear operators between complex Hilbert spaces H and K, and zt is the transpose of z induced by a conjugation on H. Cartan factors of type 2 and 3 are subtriples of B(H,H), the latter notation is shortened to B(H). The type 3 and 4 are Jordan algebras with the usual Jordan product x◦y = 1(xy+yx). A spin factor is 2 a Banach space that is equipped with a complete inner product h·,·i and a conjugation j on the resulting Hilbert space, with triple product 1 {x,y,z} = (hx,yiz +hz,yix−hx,jzijy) 2 such that the given norm and the Hilbert space norm are equivalent. By Horn’s result in [16], a JBW*-triple Z is of type I if, and only if, it is linearly isometric to an ℓ∞-sum L∞(Ω ) ⊗ C where C is a Cartan α α α α L 7 factor. Such a type I JBW*-triple will be called type I if each Cartan fin factor C is ﬁnite-dimensional. It has been shown in [7] that a JBW*-triple α Z is type I if, and only if, its predual Z has the Dunford-Pettis property. fin ∗ We recall that a Banach space W has the Dunford-Pettis property if every weakly compact operator on W is completely continuous. Such property is inherited by complemented subspaces. Horn’s type I structural result above also shows that a JBW*-algebra is type I as a JBW*-triple if and only if its self-adjoint part is a type I JBW-algebra in the sense of [15]. Lemma 1.1 Let Z be a JBW*-subtriple of a type I JBW*-triple. Then fin Z is type I . fin Proof. By [7, Corollary 6], Z has the Dunford-Pettis property. 2 ∗ We will begin our investigation of contractive projections in the next section. Acontractive projection P : Z −→ Z ona JB*-tripleZ isa bounded linear map such that P2 = P and kPk ≤ 1. We will exclude the trivial case of P = 0 which then implies kPk = 1. Given such a contractive projection P on Z with triple product {·,·,·,}, one can show, using the holomorphic characterization of JB*-triples [21, 27], that the range P(Z) is also a JB*- triple in the triple product [x,y,z] = P{x,y,z} (x,y,z ∈ P(Z)). Moreover, one has the following conditional expectation formula: P{Px,Py,Pz}= P{Px,y,Pz} (x,y,z ∈ Z). The above result has also been proved in [12] for subtriples of C*-algebras, via an operator algebra approach which also yields the formula P{Px,Py,Pz} = P{x,Py,Pz}. A weak* continuous projection on a JBW*-triple is called normal. 2 Contractive projections on JW∗-algebras In this section, we consider a JW*-algebra A ⊂ B(H) with positive part A+, inheriting various topologies of B(H). A positive linear functional ϕ 8 of A is called a trace if ϕ(sxs) = ϕ(x) for all symmetries s ∈ A and all x ∈ A+, where a symmetry in A is a self-adjoint element s such that s2 is the identity in A. By [1], every normal trace on A can be extended to a normal trace on its enveloping von Neumann algebra. Further, if ϕ is faithful, so is its extension. In the sequel, our JW∗-subalgebras need not have the same identity element as the JW∗-algebras which contain them. The following lemma is a special case of Lemma 1.1, but the proof below is intrinsic without using the Dunford-Pettis property. Lemma 2.1 Every JW*-subalgebra of a type I JW*-algebra is of type I . fin fin Proof. Let A be a JW*-subalgebra of a type I JW*-algebra B. Then A fin is ﬁnite since it is a subalgebra of a ﬁnite algebra. Let p ∈ A be a projec- tion. Then pAp is a subalgebra of the type I algebra pBp. Suppose, for fin contradiction, that pAp contains no abelian projection. By cutting down to a homogeneous summand, we may assume that pBp is homogeneous. Then by [32, Theorem 17], p can be decomposed into any number of mutually orthogonal and strongly equivalent projections in pAp. Since equivalent pro- jections in pAp are also equivalent in pBp, and since in a homogeneous type I algebra, there are at most a ﬁxed number of mutually orthogonal and fin strongly equivalent projcetions, we have a contradiction. So pAp contains an abelian projection and A is type I . 2 fin Lemma 2.2 Let (A,◦) be a JW*-algebra with identity 1 and let P : A −→ A be a contractive projection. If P(A) contains a unitary tripotent u of P(A), then P1 = P(uu∗u∗u) = P(u ◦ u∗). In addition, if u is a projection in A, then P1 = u. Proof. Recall that the triple product in P(A) is given by [x,y,z] = P{x,y,z}. Since u is a unitary tripotent in P(A), we have by the main identity P1 = [P1,u,[u,u,u]] = [[P1,u,u],u,u]−[u,[u,P1,u],u]+[u,u,[P1,u,u]] = P1−[u,[u,P1,u],u]+P1 9 and by the conditional expectation formula, P1 = [u,[u,P1,u],u] = P{u,P{u,P1,u},u} = P{u,P(u2),u} = P{u,u2,u} = P(uu∗u∗u). Also, P1 = [u,u,P1] = P{u,u,P1}= P{u,u,1} = P(u◦u∗). 2 Remark. The above result shows that there is at most one unitary tripotent in P(A) which is a projection in A. If P(A) is a JW*-subalgebra of A, then the identity in P(A) is a projection in A and P(1 ) = 1 . A P(A) As in[32], A is said to be modularif its projections forma modular lattice in which case A admits a centre-valued trace, and therefore a separating fam- ily of normal traces. It has been shown in [1] that a JW*-algebra is modular if, and only if, its enveloping von Neumann algebra is ﬁnite. For this reason, we propose from now on to replace the term “modular” by the more common term “ﬁnite” throughout. A projection p in a JW*-algebra A is called ﬁnite if the JW*-algebra pAp is ﬁnite. We recall that, for a net (x ) in a von Neumann algebra, we have α x −→ 0 strongly ⇔ x∗x −→ 0 weakly α α α x −→ 0 strongly∗ ⇔ x∗x +x x∗ −→ 0 weakly. α α α α α Plainly, strong* convergence implies strong convergence. Lemma 2.3 Let A be a JW*-algebra. The following conditions are equiva- lent: (i) A is ﬁnite. (ii) The map x ∈ A 7→ x∗ ∈ A is strongly continuous on bounded spheres in the enveloping von Neumann algebra of A. Proof. (i) ⇒ (ii). Let F be a separating family of normal traces of A. Then the family F˜ = {ϕ˜ : ϕ ∈ F} of normal tracial extensions of traces in F is separating on the enveloping von Neumann algebra A of A (cf. [1, 10

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.