Amplification of Completely Bounded Operators and Tomiyama's Slice

Ampliﬁcation of Completely Bounded Operators and Tomiyama’s Slice Maps

Matthias Neufang

Abstract Let ( , ) be a pair of von Neumann algebras, or of dual operator spaces with at least one M N of them having property Sσ, and let Φ be an arbitrary completely bounded mapping on . We present an explicit construction of an amplification of Φ to a completely bounded mappingM on . Our approach is based on the concept of slice maps as introduced by Tomiyama, and makesM⊗N use of the description of the predual of given by Effros and Ruan in terms of the operator space projective tensor product (cf. [Eff–RuaM⊗N 90], [Rua 92]). We further discuss several properties of an amplification in connection with the investigations made in [May–Neu–Wit 89], where the special case = ( ) and = ( ) has been considered (for Hilbert spaces and ). We will then mainlyM focusB H on variousN applications,B K such H K as a remarkable purely algebraic characterization of w∗-continuity using amplifications, as well as a generalization of the so-called Ge–Kadison Lemma (in connection with the uniqueness problem of amplifications). Finally, our study will enable us to show that the essential assertion of the main result in [May–Neu–Wit 89] concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama’s slice maps.

1 Introduction

The initial aim of this article is to present an explicit construction of the amplification of an arbitrary completely bounded mapping on a von Neumann algebra to a completely bounded mapping on M , where denotes another von Neumann algebra, and to give various applications. M⊗N N Of course, an amplification of a completely bounded mapping on a von Neumann algebra is easily obtained if the latter is assumed to be normal (i.e., w∗-w∗-continuous); cf., e.g., [deC–Haa 85], Lemma 1.5 (b). Our point is that we are dealing with not necessarily normal mappings and nevertheless even come up with an explicit formula for an amplification. We further remark that, of course, as already established by Tomiyama, it is possible to amplify completely positive mappings on von Neumann algebras ([Str 81], Prop. 9.4). One could then think of Wittstock’s decomposition theorem to write an arbitrary completely bounded mapping on the von Neumann algebra as a sum of four completely positive ones and apply Tomiyama’s result – M but the use of the decomposition theorem requires to be injective. Moreover, we wish to stress M 2000 Mathematics Subject Classification: 46L06, 46L07, 46L10, 47L10, 47L25. Key words and phrases: completely bounded operator, amplification, Tomiyama’s slice maps, von Neumann algebra, dual operator space, property Sσ, projective operator space tensor product. The author is currently a PIMS Postdoctoral Fellow at the University of Alberta, Edmonton, where this work was accomplished. The support of PIMS is gratefully acknowledged.

1 that this procedure would be highly non-constructive: first, the proof of Tomiyama’s result uses Banach limits in an essential way, and the decomposition theorem as well is an abstract existence result. We even go on further to show that the same construction can actually be carried out in case M and are only required to be dual operator spaces, where at least one of them shares property Sσ N (the latter is satisfied, e.g., by all semidiscrete von Neumann algebras). In this abstract situation, we do not even have a version of the above mentioned result of Tomiyama at hand. Furthermore, our approach yields, in particular, a new and elegant construction of the usual amplification of an operator Φ ( ( )) to an operator in ( ( 2 )) = ( ( ) ( )), ∈ CB B H CB B H ⊗ K CB B H ⊗B K i.e., of the mapping ( ) Φ Φ ∞ 7→ introduced in [Eff–Kis 87] (p. 265) and studied in detail in [May–Neu–Wit 89] (cf. also [Eff–Rua 88], p. 151 and Thm. 4.2, where instead of ( ), an arbitrary dual operator space B H is considered as the first factor). But of course, the main interest of our approach lies in the fact that it yields an explicit and constructive description of the amplification in a much more general setting – where the second factor ( ) is replaced by an arbitrary von Neumann algebra or even B K a dual operator space. In this case, the very definition of the mapping Φ Φ( ) does not make 7→ ∞ sense, since it is based on the representation of the elements in ( ) ( ) as infinite matrices with B H ⊗B K entries from ( ). B H The crucial idea in our construction is to use the concept of slice maps as introduced by Tomiyama, in connection with the explicit description of the predual ( ) , given by Effros M⊗N ∗ and Ruan – cf. [Eff–Rua 90], for the case of von Neumann algebras, and [Rua 92], for dual operator spaces with the first factor enjoying property Sσ. In these two cases, one has a canonical complete isometry

( ) =cb , (1) M⊗N ∗ M∗⊗N∗ b where the latter denotes the projective operator space tensor product. The dichotomic nature of our statements precisely arises from this fact; at this point, we wish to emphasize that, as shown in [Rua 92], Cor. 3.7, for a dual operator space , (1) holding for all dual operator spaces , is M N actually equivalent to having property Sσ. M Hence, the combination of operator space theory and the classical operator algebraic tool provided by Tomiyama’s slice maps enables us to show that the latter actually encodes all the essential properties of an ampliﬁcation. Summarizing the major advantages of our approach to the ampliﬁ- cation problem, we stress the following:

(a) The use of Tomiyama’s slice maps gives rise to an explicit formula for the ampliﬁcation of arbitrary completely bounded mappings with a simple structure.

(b) In particular, the approach is constructive.

(c) In various cases, it is easier to handle ampliﬁcations using our formula, since the construction does not involve any (w∗-)limits. As we shall see, the investigation of the ampliﬁcation mapping is thus mainly reduced to purely algebraic considerations – in contrast to the somehow delicate analysis used in [Eff–Rua 88] or [May–Neu–Wit 89].

2 (d) Our framework – especially concerning the class of objects allowed as “second factors” of ampliﬁcations – is by far more general than what can be found in the literature.

We would finally like to point out that, in view of the above mentioned characterization of dual operator spaces fulfilling equation (1), via property Sσ ([Rua 92], Cor. 3.7), our approach of the amplification problem seems “best possible” among the approaches satisfying (a)–(d). The paper is organized as follows. – First, we provide the necessary terminological background from the theory of operator algebras and operator spaces. We further give the exact definition of what we understand by an “amplification”. The construction of the latter in our general situation is presented in section 3. Section 4 contains a discussion of this amplification mapping under various aspects, e.g., relating it to results from [deC–Haa 85] and [May–Neu–Wit 89]. We derive a stability result concerning the passage to von Neumann subalgebras or dual operator subspaces, respectively. Furthermore, an alternative (non-constructive) description of our amplification mapping is given, which, in particular, immediately implies that the latter preserves complete positivity. In section 5, we introduce an analogous concept of “amplification in the first variable” which leads to an interesting comparison with the usual tensor product of operators on Hilbert spaces. The new phenomenon we encounter here is that the amplifications, one in the first, the other in the second variable, of two completely bounded mappings Φ ( ) and Ψ ( ) do not ∈ CB M ∈ CB N commute in general. They do, though, if Φ or Ψ happens to be normal (cf. Theorem 5.1 below). But our main result of this section, which may even seem a little estonishing, states that for most von Neumann algebras and , the converse holds true. This means that the apparently weak M N – and purely algebraic! – condition of commutation of the amplification of Φ with every amplified Ψ already suffices to ensure that Φ is normal (Theorem 5.4). The proof of the latter assertion is rather technical and involves the use of countable spiral nebulas, a concept first introduced by Hofmeier–Wittstock in [Hof–Wit 97]. In the section thereafter, we turn to the uniqueness problem of amplifications. Of course, an amplification which only meets the obvious algebraic condition is far from being unique; but we shall show that with some (weak) natural assumptions, one can indeed establish uniqueness. One of our two positive results actually yields a considerable generalization (Proposition 6.2) of a theorem which has proved useful in the solution of the splitting problem for von Neumann algebras (cf. [Ge–Kad 96], [Str–Zsi 99]), and which is sometimes referred to as the Ge–Kadison Lemma. Finally, section 7 is devoted to a completely new approach to a theorem of May, Neuhardt and Wittstock regarding a module homomorphism property of the amplified mapping. We prove a version of their result in our context, which reveals that the essential assertion in fact relies on a fundamental and elementary module homomorphism property shared by Tomiyama’s slice maps.

2 Preliminaries and basic deﬁnitions

Since a detailed account of the theory of operator spaces, as developed by Blecher–Paulsen, Effros–Ruan, Pisier et al., is by now available via different sources ([Eff–Rua 00], [Pis 00], [Wit et al. 99]), as for its basic facts, we shall restrict ourselves to fix the terminology and notation we use.

3 Let X and Y be operator spaces. We denote by (X,Y ) the operator space of completely CB bounded maps from X to Y , endowed with the completely bounded norm cb. We further write k · k (X) for (X,X). If and are von Neumann algebras with , then we denote by CB CB M R R ⊆ M ( ) the space of all completely bounded -bimodule homomorphisms on . If two operator CBR M R M spaces X and Y are completely isometric, then we write X =cb Y . An operator space Y is called a dual operator space if there is an operator space X such that Y is completely isometric to X∗. In this case, X is called an operator predual, or predual, for short, of the operator space Y . In general, the predual of a dual operator space is not unique (up to complete isometry); see [Rua 92], p. 180, for an easy example. For a dual operator space Y with a given predual X, we denote by σ(Y ) the subspace of CB (Y ) consisting of normal (i.e., σ(Y,X)-σ(Y,X)-continuous) mappings. CB If is a Hilbert space, then, of course, every w -closed subspace Y of ( ) is a dual operator H ∗ B H space with a canonical predual ( ) /Y , where Y denotes the preannihilator of Y in ( ) . B H ∗ ⊥ ⊥ B H ∗ We denote this canonical predual by Y . Conversely, if Y is a dual operator space with a given ∗ predual X, there is a w∗-homeomorphic complete isometry of Y onto a w∗-closed subspace Y of ( ), for some suitable Hilbert space (see [Eff–Rua 90], Prop. 5.1). Following the terminologye B H H of [Eff–Kra–Rua 93], p. 3, we shall call such a map a w∗-embedding. In this case, X is completely isometric to the canonical predual Y , and we shall identify Y with Y (cf. the discussion in [Rua 92], ∗ p. 180). e e We denote by 2 the Hilbert space tensor product of two Hilbert spaces and . For H ⊗ K H K operator spaces X and Y , we write X ∨ Y and X Y for the injective and projective operator space ⊗ ⊗ tensor product, respectively. We recall that, forb operator spaces X and Y , we have a canonical complete isometry: cb (X Y )∗ = (X,Y ∗). ⊗ CB If and are von Neumann algebras,b we denote as usual by the von Neumann algebra M N M⊗N tensor product. More generally, let V ∗ and W ∗ be dual operator spaces with given w∗-embeddings V ( ) and W ( ). Then the w -spatial tensor product of V and W , still denoted by ∗ ⊆ B H ∗ ⊆ B K ∗ ∗ ∗ V W , is deﬁned to be the w -closure of the algebraic tensor product V W in ( 2 ). The ∗⊗ ∗ ∗ ∗ ⊗ ∗ B H ⊗ K corresponding w∗-embedding of V ∗ W ∗ determines a predual (V ∗ W ∗) . Since we have ⊗ ⊗ ∗ cb (V ∗ W ∗) = V nuc W ⊗ ∗ ⊗ completely isometrically (where nuc denotes the nuclear operator space tensor product), the w - ⊗ ∗ spatial tensor product of dual operator spaces does not depend on the given w∗-embedding (cf. [Eff–Kra–Rua 93], p. 127). In the following, whenever we are speaking of “a dual operator space”, we always mean “a dual operator space with a given (operator) predual, denoted by ”. M M∗ We will now present, in the general framework of dual operator spaces, the construction of left and right slice maps, following the original concept ﬁrst introduced by Tomiyama for von Neumann algebras ([Tom 70], p. 4; cf. also [Kra 91], p. 119). – Let and be dual operator spaces. For M N any τ and any u , the mapping ∈ N∗ ∈ M⊗N ρ u, ρ τ M∗ 3 7→ h ⊗ i

4 is a continuous linear functional on , hence deﬁnes an element Lτ (u) of . As is easily veriﬁed, M∗ M Lτ is the unique w -continuous linear map from to such that ∗ M⊗N M

Lτ (S T ) = τ, T S (S ,T ); ⊗ h i ∈ M ∈ N it will be called the left slice map associated with τ. In a completely analogous fashion, we see that for every ρ , there is a unique w∗-continuous linear map Rρ from to such that ∈ M∗ M⊗N N

Rρ(S T ) = ρ, S T (S ,T ), ⊗ h i ∈ M ∈ N which we call the right slice map associated with ρ. We note that, as an immediate consequence, we have for every ρ , τ and u : ∈ M∗ ∈ N∗ ∈ M⊗N

Lτ (u), ρ = u, ρ τ and Rρ(u), τ = u, ρ τ . h i h ⊗ i h i h ⊗ i

Let us now briefly recall the definition of property Sσ for dual operator spaces, as introduced by Kraus ([Kra 83], Def. 1.4). To this end, consider two dual operator spaces and with w - M N ∗ embeddings ( ) and ( ). Then the Fubini product of and with respect to M ⊆ B H N ⊆ B K M N ( ) and ( ) is defined through B H B K

( , , ( ), ( )) u ( 2 ) Lτ (u) ,Rρ(u) for all ρ ( ) , τ ( ) . F M N B H B K { ∈ B H ⊗ K | ∈ M ∈ N ∈ B H ∗ ∈ B K ∗} p By the important result [Rua 92], Prop. 3.3, we have

cb ( , , ( ), ( )) = ( )∗, F M N B H B K M∗⊗N∗ b which in particular shows that the Fubini product actually does not depend on the particular choice of Hilbert spaces and (cf. also [Kra 83], Remark 1.2). Hence, we may denote the Fubini H K product simply by ( , ). Now, a dual operator space is said to have property Sσ if F M N M ( , ) =cb F M N M⊗N holds for every dual operator space . N The question of whether every von Neumann algebra has property Sσ has been answered in the negative by Kraus ([Kra 91], Thm. 3.3); in fact, there exist even separably acting factors without property Sσ ([Kra 91], Thm. 3.12). But every semidiscrete von Neumann algebra – and hence, every type I von Neumann algebra ([Eff–Lan 77], Prop. 3.5) – has property Sσ, as shown by Kraus in [Kra 83], Thm. 1.9. We ﬁnally refer to the very interesting results obtained by Eﬀros, Kraus and Ruan showing that property Sσ is actually equivalent to various operator space approximation properties (cf. [Kra 91], Thm. 2.6; [Eff–Kra–Rua 93], Thm. 2.4). To avoid long paraphrasing, let us now introduce some terminology suitable for the statement of our results.

Deﬁnition 2.1. Let and be either M N (i) arbitrary von Neumann algebras,

5 or

(ii) dual operator spaces such that at least one of them has property Sσ. Then, in both cases, we call ( , ) an admissible pair. M N If ( , ) is an admissible pair, then we have a canonical complete isometry: M N ( ) =cb . M⊗N ∗ M∗⊗N∗ b In case and are both von Neumann algebras or dual operator spaces with having property M N M Sσ, this follows from [Rua 92], Cor. 3.6 and Cor. 3.7, respectively. If and are dual operator spaces with enjoying property Sσ, the above identiﬁcation M N N is seen as follows. (We remark that the argument below, together with [Rua 92], Cor. 3.7, shows that property Sσ of a dual operator space – resp. – is actually characterized by the equality M N ( ) =cb holding for all dual operator spaces – resp. .) M⊗N ∗ M∗⊗N∗ N M Let V be anb arbitrary operator space. Then, by [Kra 91], Thm. 2.6, V ∗ has property Sσ if and only if it has the weak∗ operator approximation property (in the terminology of [Eff–Kra–Rua 93]). Owing to [Eff–Kra–Rua 93], Thm. 2.4 (6), this is equivalent to

cb V W = V nuc W ⊗ ⊗ b holding for all operator spaces W . But since both the projective and the injective operator space tensor product are symmetric, this in turn is equivalent to the canonical mapping

Φ: W V W ∨ V ⊗ −→ ⊗ b being one-to-one for all operator spaces W . But this is the case if and only if we have

cb W nuc V = W V ⊗ ⊗ b for all operator spaces W , which ﬁnally is of course equivalent to

cb (W ∗ V ∗) = W V ⊗ ∗ ⊗ b holding for all operator spaces W . We shall now make precise what we mean by an ampliﬁcation, in particular, which topological requirements it should meet beyond the obvious algebraic ones.

Deﬁnition 2.2. Let ( , ) be an admissible pair. A completely bounded linear mapping M N χ : ( ) ( ), CB M −→ CB M⊗N satisfying the algebraic ampliﬁcation condition (AAC)

χ(Φ)(S T ) = Φ(S) T ⊗ ⊗ for all Φ ( ), S , T , will be called an ampliﬁcation, if in addition it enjoys the ∈ CB M ∈ M ∈ N following properties:

6 (i) χ is a complete isometry,

(ii) χ is multiplicative (hence, by (i), an isometric algebra isomorphism onto the image),

(iii) χ is w∗-w∗-continuous, (iv) χ( σ( )) σ( ), i.e., normality of the mapping is preserved. CB M ⊆ CB M⊗N Remark 2.3. In particular, χ is unital. For χ(id ) coincides with id on all elementary tensors M S T due to (AAC); furthermore, by (iv), it is normal, whenceM⊗N we obtain χ(id ) = id ⊗ ∈ M⊗N M M⊗N on the whole space . M⊗N In [May–Neu–Wit 89] – cf. also [Eff–Kis 87], p. 265 –, for every Φ ( ( )), the au- ( ) ∈ CB B H thors obtain a mapping Φ ∞ ( ( 2 )) = ( ( ) ( )) which obviously fulfills ∈ CB B H ⊗ K CB B H ⊗B( )K the algebraic amplification condition (AAC). The definition of Φ ∞ is carried out in detail in [May–Neu–Wit 89], p. 284, 2 (there, the Hilbert space is written in the form `2(I) for some § K suitable set I). We restrict ourselves to recall that, given a representation of u ( 2 ) by an ∈ B H ⊗ K infinite matrix [ui,j]i,j, where ui,j ( ), one has: ∈ B H ( ) Φ ∞ ([ui,j]i,j) = [Φ(ui,j)]i,j. In fact, as is well-known, more than only (AAC) is satisfied. Indeed, we have Theorem 2.4. The mapping

( ( )) ( ( ) ( )) CB B H −→ CB( B) H ⊗B K Φ Φ ∞ 7→ fulfills the above requirements of an amplification. Proof. This follows from [May–Neu–Wit 89], p. 284, 2 and Prop. 2.2; condition (iv) in Definition § 2.2 is shown in [Hof 95], Satz 3.1.

We close this section by remarking that several properties of the mapping Φ Φ( ) are studied 7→ ∞ in a more general situation by Eﬀros and Ruan in [Eff–Rua 88] (see p. 151 and Thm. 4.2).

3 The main construction

We now come to our central result concerning the existence of an amplification in the sense of Definition 2.2 and, which is most important, presenting its explicit form. Theorem 3.1. Let ( , ) be an admissible pair. Then there exists an amplification M N χ : ( ) ( ) CB M −→ CB M⊗N in the sense of Definition 2.2. The amplification is explicitly given by

χ(Φ)(u), ρ τ = Φ(Lτ (u)), ρ , h ⊗ i h i where Φ ( ), u , ρ , τ . – Here, Lτ denotes the left slice map associated ∈ CB M ∈ M⊗N ∈ M∗ ∈ N∗ with τ.

7 Proof. We construct a complete quotient mapping

κ : ( ) ( ) CB M⊗N ∗ −→ CB M ∗ in such a way that χ κ∗ will enjoy the desired properties. We ﬁrst note thatp

( ) =cb (2) CB M ∗ M∗⊗M b and, analogously,

( ) =cb ( ) ( ) (3) CB M⊗N ∗ M⊗N ∗⊗ M⊗N b with completely isometric identiﬁcations. We further write W : ( , ( , )) ( ( ), ) CB M⊗N CB N∗ M −→ CB N∗⊗ M⊗N M for the canonical complete isometry. For τ , let us considerb the left slice map ∈ N∗

Lτ : , M⊗N −→ M where Lτ (S T ) = τ, T S ⊗ h i for S , T . In [Rua 92], Thm. 3.4 (and Cor. 3.6, 3.7), Ruan constructs a complete isometric ∈ M ∈ N isomorphism (cf. also [Eff–Rua 90], Thm. 3.2)

θ : ( ) , M∗⊗N∗ −→ M⊗N ∗ b where for elementary tensors we have (S , T , ρ , τ ): ∈ M ∈ N ∈ M∗ ∈ N∗ θ(ρ τ),S T = ρ, S τ, T . (4) h ⊗ ⊗ i h ih i Thus for all ρ and τ , we see that θ(ρ τ) = ρ τ on the algebraic tensor product ∈ M∗ ∈ N∗ ⊗ ⊗ ; but since θ(ρ τ), ρ τ ( ) are normal functionals, on the whole space we M ⊗ N ⊗ ⊗ ∈ M⊗N ∗ M⊗N obtain for all ρ , τ : ∈ M∗ ∈ N∗ θ(ρ τ) = ρ τ. (5) ⊗ ⊗ Hence, in the sequel, we shall not explicitly note θ when applied on elementary tensors, except for special emphasis. We now obtain a complete isometry:

θ∗ : ( , ). M⊗N −→ CB N∗ M For each τ , we deﬁne a linear continuous mapping ∈ N∗

ϕτ : M⊗N −→ M through ϕτ (u) θ∗(u)(τ) p 8 for u . One immediately deduces from the deﬁnition that ϕτ is normal. Furthermore, ϕτ ∈ M⊗N satisﬁes ϕτ (S T ) = τ, T S ⊗ h i for all S , T . – For if ρ , we have: ∈ M ∈ N ∈ M∗

ϕτ (S T ), ρ = θ∗(S T )(τ), ρ h ⊗ i h ⊗ i = θ∗(S T ), ρ τ h ⊗ ⊗ i = θ(ρ τ),S T h ⊗ ⊗ i (4) = ρ, S τ, T . h ih i

Hence, ϕτ coincides with the left slice map Lτ on all elementary tensors in , and since both M⊗N mappings are normal, we ﬁnally see that ϕτ = Lτ . Thus we obtain

θ∗(u)(τ) = Lτ (u) for all u , τ . We note that, of course, ∈ M⊗N ∈ N∗

θ∗ ( , ( , )). ∈ CB M⊗N CB N∗ M Hence, by deﬁnition of W , we have:

W (θ∗) ( ( ), ). ∈ CB N∗⊗ M⊗N M b Thanks to the functorial property of the projective operator space tensor product, we now get a completely bounded mapping

id W (θ∗): ( ) . M∗ ⊗ M∗⊗N∗⊗ M⊗N −→ M∗⊗M b b b b What remains to be done, is the passage from ( ) ( ) to ( ). This is M⊗N ∗⊗ M⊗N M∗⊗N∗⊗ M⊗N accomplished by the complete quotient mapping b b b

1 θ− id :( ) ( ) ( ). ⊗ M⊗N M⊗N ∗⊗ M⊗N −→ M∗⊗N∗⊗ M⊗N b b b b Now, deﬁning

1 κ (id W (θ∗)) (θ− id ):( ) ( ) , M∗ ⊗ ◦ ⊗ M⊗N M⊗N ∗⊗ M⊗N −→ M∗⊗M p b b b b we obtain a completely bounded mapping which has the desired properties, as we shall now prove. In view of (2) and (3), we see that we have indeed constructed a completely bounded mapping

κ : ( ) ( ) , CB M⊗N ∗ −→ CB M ∗ and it remains to verify the properties of the mapping

χ κ∗ : ( ) ( ). CB M −→ CB M⊗N p First, we remark the following:

9 ( ) Since the mapping θ : ( ) is a complete isometry, it is read- ∗ M∗⊗N∗ −→ M⊗N ∗ ily seen that θ( ) ( ) isb norm dense in ( ) ( ). Every element of M∗⊗Ncb∗ ⊗ M⊗N M⊗N ∗⊗ M⊗N the space ( ) = (( ) ( ))∗ is thus completelyb determined by its values on CB M⊗N M⊗N ∗⊗ M⊗N θ( ) ( ). b M∗⊗N∗ ⊗ M⊗N By construction, we see that for all Φ ( ), u , ρ and τ : ∈ CB M ∈ M⊗N ∈ M∗ ∈ N∗ χ(Φ), θ(ρ τ) u = Φ, κ(θ(ρ τ) u) h ⊗ ⊗ i h ⊗ ⊗ i = Φ, ρ W (θ∗)(τ u) h ⊗ ⊗ i = Φ, ρ θ∗(u)(τ) h ⊗ i = Φ, ρ Lτ (u) . h ⊗ i Hence we have:

χ(Φ), θ(ρ τ) u = Φ, ρ Lτ (u) , h ⊗ ⊗ i h ⊗ i and following ( ), by this χ(Φ) is completely determined. Taking into account (5), we thus have ∗ that for all ρ , τ , u : ∈ M∗ ∈ N∗ ∈ M⊗N

χ(Φ), ρ τ u = Φ, ρ Lτ (u) , (6) h ⊗ ⊗ i h ⊗ i which determines χ(Φ) completely. Now, let us verify the condition (AAC) and properties (i)–(iv). (AAC) Fix Φ ( ), S , T , ρ and τ . We obtain: ∈ CB M ∈ M ∈ N ∈ M∗ ∈ N∗ ρ τ, χ(Φ)(S T ) = χ(Φ), ρ τ (S T ) h ⊗ ⊗ i h ⊗ ⊗ ⊗ i (6) = Φ, ρ Lτ (S T ) h ⊗ ⊗ i = τ, T Φ, ρ S h ih ⊗ i = ρ, Φ(S) τ, T h ih i = ρ τ, Φ(S) T . h ⊗ ⊗ i Hence, for all Φ ( ), S and T , we have: ∈ CB M ∈ M ∈ N χ(Φ)(S T ) = Φ(S) T, ⊗ ⊗ as desired. (i) To prove that χ is a complete isometry, we show that κ = χ is a complete quotient mapping. ∗ But this follows from the fact that W (θ∗) is a complete quotient mapping. (For if this is established, then the mapping id W (θ∗) ( ( ), ) is a complete quotient map, and M∗ ⊗ ∈ CB1 M∗⊗N∗⊗ M⊗N M∗⊗M of course this is also theb case for θ− id b b (( ) b( ), ( )).) This ⊗ M⊗N ∈ CB M⊗N ∗⊗ M⊗N M∗⊗N∗⊗ M⊗N in turn is easily seen. Fix n N.b Let S = [Si,j]i,j Mn( b ), S < 1.b We areb looking for an ∈ ∈ M(n) k k (n) element ϕ Mn( ( )), such that ϕ < 1 and W (θ∗) (ϕ) = S, where W (θ∗) denotes ∈ N∗⊗ M⊗N k k the nth ampliﬁcationb of the mapping W (θ∗).

10 Let us choose τ with τ = 1. Then there exists T , T = 1, such that τ, T = 1. ∈ N∗ k k ∈ N k k h i Now ϕ [τ (Si,j T )]i,j Mn( ( )) satisﬁes all our requirements. For we have: ⊗ ⊗ ∈ N∗⊗ M⊗N p b ϕ = τ [S T ] Mn( ( )) i,j i,j Mn( ) k k N∗⊗ M⊗N k k k ⊗ k M⊗N b = τ [Si,j T ]i,j k k k ⊗ kMn( ∨ ) M⊗N = τ [Si,j]i,j Mn( ) T k k k k M k k < 1.

Furthermore, we see that:

(n) (n) W (θ∗) (ϕ) = W (θ∗) ([τ (Si,j T )]i,j) ⊗ ⊗ = [W (θ∗)(τ (Si,j T ))]i,j ⊗ ⊗ = [θ∗(Si,j T )(τ)]i,j ⊗ = [Lτ (Si,j T )]i,j ⊗ = [ τ, T Si,j]i,j h i = S, which establishes the claim. (ii) We show that χ is an algebra homomorphism. – To this end, let Φ, Ψ ( ). We have ∈ CB M to show that χ(ΦΨ) = χ(Φ)χ(Ψ), as elements in cb ( ) = ( ( ))∗. CB M⊗N M∗⊗N∗⊗ M⊗N Fix T , ρ and τ . It suﬃces tob showb that ∈ M⊗N ∈ M∗ ∈ N∗ χ(ΦΨ), ρ τ T = χ(Φ)χ(Ψ), ρ τ T . h ⊗ ⊗ i h ⊗ ⊗ i On the left side, we obtain:

(6) χ(ΦΨ), ρ τ T = ΦΨ, ρ Lτ (T ) h ⊗ ⊗ i h ⊗ i = Φ(Ψ(Lτ (T ))), ρ h i = Φ, ρ Ψ(Lτ (T )) . h ⊗ i On the right, we have:

χ(Φ)χ(Ψ), ρ τ T = χ(Φ)[χ(Ψ)(T )], ρ τ h ⊗ ⊗ i h ⊗ i = χ(Φ), ρ τ χ(Ψ)(T ) h ⊗ ⊗ i (6) = Φ, ρ Lτ (χ(Ψ)(T )) . h ⊗ i Thus we have to show that Lτ (χ(Ψ)(T )) = Ψ(Lτ (T )),

11 as elements in . To this end, let ρ . We then have: M ∈ M∗

Lτ (χ(Ψ)(T )), ρ = χ(Ψ)(T ), ρ τ h i h ⊗ i = χ(Ψ), ρ τ T h ⊗ ⊗ i (6) = Ψ, ρ Lτ (T ) h ⊗ i = Ψ(Lτ (T )), ρ , h i which yields the desired equality. (iii) Being an adjoint mapping, χ is clearly w∗-w∗-continuous. σ σ (iv) Fix Φ ( ). We wish to prove that χ(Φ) ( ). So let (Tα) Ball( ) ∈ CB Mw∗ ∈ CB M⊗N ⊆ M⊗N be a net such that Tα 0. We claim that χ(Φ)(Tα) 0 (σ( , )). It suﬃces to −→ −→ M⊗N M∗⊗N∗ show that we have for arbitrary ρ and τ : b ∈ M∗ ∈ N∗

χ(Φ)(Tα), ρ τ 0. h ⊗ i −→ We obtain for all indices α:

χ(Φ)(Tα), ρ τ = χ(Φ), ρ τ Tα h ⊗ i h ⊗ ⊗ i (6) = Φ, ρ Lτ (Tα) h ⊗ i = Φ(Lτ (Tα)), ρ . h i

w∗ Since Φ is normal, it thus remains to show that Lτ (Tα) 0; but this follows in turn from the −→ normality of the slice map Lτ .

4 Discussion of the ampliﬁcation mapping

In the sequel, for a mapping Φ ( ), we shall consider amplifications with respect to different ∈ CB M von Neumann algebras or dual operator spaces . We will show that all of these amplifications are N compatible with each other in a very natural way. For this purpose, we introduce the following

Notation 4.1. If ( , ) is an admissible pair, we will denote by χ the amplification with respect M N N to as constructed in Theorem 3.1. N Taking the restriction of the mapping χ : ( ) ( ) to the subalgebra σ( ), N CB M −→ CB M⊗Nσ σ CB M by condition (iv) in Definition 2.2, we obtain a mapping χ0 : ( ) ( ), which CB M −→ CB M⊗N of course satisfies the algebraic condition (AAC) on σ( ). But this is the case as well for the CB M amplification H : σ( ) σ( ) CB M −→ CB M⊗N considered, for von Neumann algebras and , by de Cannièreand Haagerup in [deC–Haa 85], M N Lemma 1.5 (b). Note that H is completely determined by (AAC), since the image of H consists of normal mappings. Hence we deduce that χ0 = H, thus showing that χ is indeed a (w∗- N w -continuous) extension of H to ( ), where and are von Neumann algebras (cf. also ∗ CB M M N [Kra 91], p. 123, for a discussion of the mapping H in a more general setting).

12 Remark 4.2. Theorem 3.1 implies in particular (cf. (iv) in Definition 2.2) that whenever ( , ) M N is an admissible pair, then for every Φ σ( ), the mapping χ (Φ) is the unique normal map in ∈ CB M N σ( ) which satisfies (AAC). This entails that the last assumption made in [Kra 83], Prop. CB M⊗N 1.19 is always fulfilled (and even true in greater generality than actually needed there), which besides slightly simplifies the proof of [Kra 83], Thm. 2.2.

Let us now briefly return to the special case where = ( ) and = ( ). We already M B H N B K noted in Theorem 2.4 that the mapping Φ Φ( ), where Φ ( ( )), is an amplification in the 7→ ∞ ∈ CB B H sense of Definition 2.2. Since ( ) trivially is an injective factor, we conclude by Proposition 6.1 B H that

( ) χ ( )(Φ) = Φ ∞ (7) B K for all Φ ( ( )). Thus, restricted to that special case, Theorem 3.1 provides a new construction ∈ CB B H of the mapping Φ Φ( ) and for the first time shows the intimate relation to Tomiyama’s slice 7→ ∞ maps. We finally stress that our construction of χ ( )(Φ) is coordinate free – in contrast to the definition ( ) B K of Φ ∞ in [May–Neu–Wit 89]: since our approach does not rely on an explicit representation of operators in ( ) ( ) as infinite matrices, we do not have to choose a basis in the Hilbert space B H ⊗B K , whereas in [May–Neu–Wit 89], the latter is represented as `2(I) for some suitable I. The K equality ( ) χ ( )(Φ) = Φ ∞ (Φ ( ( ))), B K ∈ CB B H ( ) obtained above, now immediately implies that the definition of the mapping Φ ∞ in [May–Neu–Wit 89] does not depend on the particular choice of a basis. (This fact of course can ( ) also be obtained intrinsically from the construction of Φ ∞ , but it comes for free in our approach.) We now come to the result announced above which establishes the compatibility of our amplification with respect to different von Neumann algebras or dual operator spaces , respectively. N Theorem 4.3. Let ( , ) be an admissible pair. Let further 0 be a dual operator subspace M N N ⊆ N of such that ( , 0) is admissible. Then, for all Φ ( ), we have: N M N ∈ CB M χ (Φ) = χ (Φ). 0 0 N |M⊗N N

Proof. Denote by ι : 0 , the canonical embedding, and write q : ( 0) M⊗N → M⊗N N∗ N ∗ for the restriction map. Obviously, the pre-adjoint mapping ι : ( 0) satisﬁes ∗ M∗⊗N∗ M∗⊗ N ∗ ι = id q. b b ∗ M∗ ⊗ We shallb prove that the following diagram commutes:

/ χ (Φ) M⊗NO N M⊗NO ι ι χ (Φ) ? N0 ? 0 / 0 M⊗N M⊗N This yields the desired equality (and shows at the same time that χ (Φ) indeed leaves the 0 N |M⊗N space 0 invariant). M⊗N

13 To this end, ﬁx u 0. We have to show that ∈ M⊗N

χ (Φ)(ι(u)) = ι(χ 0 (Φ)(u)). N N Let ρ and τ . It is suﬃcient to verify that ∈ M∗ ∈ N∗

χ (Φ)(ι(u)), ρ τ = ι(χ 0 (Φ)(u)), ρ τ . h N ⊗ i h N ⊗ i For the left-hand expression, we get:

χ (Φ)(ι(u)), ρ τ = χ (Φ), ρ τ ι(u) = Φ, ρ Lτ (ι(u)) . h N ⊗ i h N ⊗ ⊗ i h ⊗ i On the right-hand side, we ﬁnd:

ι(χ 0 (Φ)(u)), ρ τ = χ 0 (Φ)(u), ρ q(τ) h N ⊗ i h N ⊗ i = χ 0 (Φ), ρ q(τ) u h N ⊗ ⊗ i = Φ, ρ L (u) . h ⊗ q(τ) i It thus remains to be shown that Lτ (ι(u)) = Lq(τ)(u). But for ϕ , we obtain: ∈ M∗

Lτ (ι(u)), ϕ = ι(u), ϕ τ h i h ⊗ i = u, ϕ q(τ) h ⊗ i = L (u), ϕ , h q(τ) i which yields the claim.

We shall now present another very natural, though highly non-constructive, approach to the amplification problem, and discuss its relation to our concept. – Let ( , ) be an admissible M N pair, with ( ) and ( ). An element Φ ( ) may be considered as a completely M ⊆ B H N ⊆ B K ∈ CB M bounded mapping from with values in ( ). Hence we can assign to Φ a Wittstock-Hahn-Banach M B H extension Φ ( ( )). Then the mapping ∈ CB B H e ( ) Φ ∞ : ( ) ( ) |M⊗N M⊗N −→ B H ⊗B K e is completely bounded and of course satisfies the algebraic amplification condition (AAC). As we shall now see, this (non-constructive) abstract procedure yields indeed a mapping which takes values in and does not depend on the choice of the Wittstock-Hahn-Banach extension – namely, ( M) ⊗N Φ ∞ is nothing but χ (Φ). This will be proved in a similar fashion as Theorem 4.3. |M⊗N N e Theorem 4.4. Let ( , ) be an admissible pair, where ( ) and ( ). Let further M N M ⊆ B H N ⊆ B K Φ ( ). Then for an arbitrary Wittstock-Hahn-Banach extension Φ ( ( )) obtained as ∈ CB M ∈ CB B H above, we have: e ( ) Φ ∞ = χ (Φ). |M⊗N N e

14 Proof. We write ι : , ( ) ( ) for the canonical embedding. It is suﬃcient to show that M⊗N → B H ⊗B K the following diagram commutes:

( ) ( ) / ( ) ( ) ( ) B H ⊗BO K Φ ∞ B H ⊗BO K ι e ι ? χ (Φ) ? N / M⊗N M⊗N Fix u , ρ ( ) , τ ( ) . We have only to show that ∈ M⊗N ∈ B H ∗ ∈ B K ∗ ( ) ι[χ (Φ)(u)], ρ τ = Φ ∞ (ι(u)), ρ τ , h N ⊗ i h ⊗ i e which by equation (7) is equivalent to

Φ[Lτ (ι(u))] = Φ[ι0(Lτ (u))] = ι0[Φ(Lτ (u))]. (9) e e |N |N Now we ﬁnd for the right-hand side term of equation (8):

Remark 4.5. If ( ), ( ) are von Neumann algebras and Φ ( ) is com- M ⊆ B H N ⊆ B K ∈ CP M pletely positive, then, using an arbitrary Arveson extension of Φ to a completely positive map ( ) Φ ( ( )), we obtain by the same reasoning as above that Φ ∞ = χ (Φ). But it is ∈ CP B H ( ) |M⊗N N easye to see from the deﬁnition of the mapping Φ Φ ∞ (cf. [May–Neu–Wite 89], p. 284, 2) ( ) 7→ § that Φ ∞ still is completely positive, whence χ e(Φ) alsoe is. This shows that our ampliﬁcation χ N N preservese complete positivity.

15 5 An algebraic characterization of normality

In the following, let still ( , ) be an admissible pair. Using Tomiyama’s left slice map, for every M N completely bounded mapping Φ ( ), we have constructed an amplification χ (Φ) =: Φ id ∈ CB M N ⊗ N ∈ ( ). In an analogous fashion, replacing in the above construction Tomiyama’s left by the CB M⊗N appropriate right slice maps, we obtain – mutatis mutandis – an amplification of completely bounded mappings Ψ ( ) “in the left variable”. This yields an amplification id Ψ ( ), ∈ CB N M⊗ ∈ CB M⊗N which is completely determined by the equation:

id Ψ, ρ τ u = Ψ, τ Rρ(u) , (10) h M⊗ ⊗ ⊗ i h ⊗ i where u , ρ and τ are arbitrary. Of course, id Ψ shares analogous properties ∈ M⊗N ∈ M∗ ∈ N∗ M⊗ to those of Φ id . ⊗ N It is natural to ask for some relation between the operators in ( ) which arise by the CB M⊗N two different kinds of amplification. Let us briefly compare our situation with the classical setting of amplification of operators on Hilbert spaces and . For operators S ( ) and T ( ), H K ∈ B H ∈ B K we consider the amplifications S id and id T in ( ) ( ) = ( 2 ). Then it is evident ⊗ K H⊗ B H ⊗B K B H⊗ K that we have

(S id ) (id T ) = (id T )(S id ), (11) ⊗ K H⊗ H⊗ ⊗ K and the tensor product of S and T is defined precisely to be this operator. Back in our situation, for Φ ( ) and Ψ ( ), the corresponding amplifications Φ id ∈ CB M ∈ CB N ⊗ N and id Ψ will commute provided at least one of the involved mappings Φ or Ψ is normal – but not M⊗ in general. In fact, we will even show that for most von Neumann algebras and ,Φ ( ) M N ∈ CB M satisfying this innocent looking commutation relation for all Ψ ( ) forces Φ to be normal! ∈ CB N This phenomenon, which may be surprising at first glance, is due to the fact that, in contrast to the multiplication in a von Neumann algebra (in our case = ), the multiplication in ( ) R R M⊗N CB R is not w -continuous in both variables, but only in the left one. In fact, the subset of ( ) ( ) ∗ CB R ×CB R on which the product is w -continuous is precisely ( ) σ( ). This is reflected in the following ∗ CB R ×CB R result which parallels relation (11) in our context.

Theorem 5.1. Let ( , ) be an admissible pair. If Ψ σ( ) is a normal mapping, then we M N ∈ CB N have for every Φ ( ): ∈ CB M (Φ id ) (id Ψ) = (id Ψ) (Φ id ). ⊗ N M⊗ M⊗ ⊗ N Proof. Fix u , ρ and τ . We ﬁrst remark that ∈ M⊗N ∈ M∗ ∈ N∗ (10) id Ψ, ρ τ u = Ψ, τ Rρ(u) h M⊗ ⊗ ⊗ i h ⊗ i = Ψ(Rρ(u)), τ h i = Rρ(u), Ψ (τ) h ∗ i = u, ρ Ψ (τ) , (12) h ⊗ ∗ i where Ψ denotes the pre-adjoint mapping of Ψ. ∗

16 To establish the claim, it suﬃces to prove that:

(Φ id ) (id Ψ), ρ τ u = (id Ψ) (Φ id ), ρ τ u . h ⊗ N M⊗ ⊗ ⊗ i h M⊗ ⊗ N ⊗ ⊗ i The left side takes on the following form:

(Φ id ) (id Ψ), ρ τ u = χ (Φ), ρ τ (id Ψ)(u) h ⊗ N M⊗ ⊗ ⊗ i h N ⊗ ⊗ M⊗ i = Φ, ρ Lτ [(id Ψ)(u)] , h ⊗ M⊗ i whereas on the right side, we obtain:

(id Ψ) (Φ id ), ρ τ u = (id Ψ) (χ (Φ)(u)), ρ τ h M⊗ ⊗ N ⊗ ⊗ i h M⊗ N ⊗ i (12) = χ (Φ)(u), ρ Ψ (τ) h N ⊗ ∗ i = χ (Φ), ρ Ψ (τ) u h N ⊗ ∗ ⊗ i = Φ, ρ LΨ (τ)(u) . h ⊗ ∗ i Hence, it remains to be shown that

Lτ [(id Ψ)(u)] = LΨ (τ)(u). M⊗ ∗

But for arbitrary ρ0 , we get: ∈ M∗

Lτ [(id Ψ)(u)], ρ0 = (id Ψ)(u), ρ0 τ h M⊗ i h M⊗ ⊗ i (12) = u, ρ0 Ψ (τ) h ⊗ ∗ i = LΨ (τ)(u), ρ0 , h ∗ i which yields the claim.

We now come to the main result of this section, which establishes a converse of the above Theorem for a very wide class of von Neumann algebras. It may be interpreted as a result on “automatic normality”, i.e., some simple, purely algebraic condition – namely commuting of the associated amplifications – automatically implies the strong topological property of normality. Besides one very mild technical condition, the first von Neumann algebra, , can be completely arbitrary, the M second one, , is only assumed to be properly infinite; we will show by an example that without the N latter condition, the conclusion does not hold in general. The weak technical assumption mentioned says, roughly speaking, that the von Neumann algebras involved should not be acting on Hilbert spaces of pathologically large dimension. We remark that for separably acting von Neumann algebras and , or, more generally, in case is countably decomposable, our assumption is trivially M N M⊗N satisfied. In order to formulate the condition precisely, we use the following natural terminology which has been introduced in [Neu 01].

Deﬁnition 5.2. Let ( ) be a von Neumann algebra. Then we deﬁne the decomposability R ⊆ B H number of , denoted by dec( ), to be the smallest cardinal number κ such that every family of R R non-zero pairwise orthogonal projections in has at most cardinality κ. R Remark 5.3. Of course, a von Neumann algebra is countably decomposable if and only if R dec( ) 0. R ≤ ℵ 17 We are now ready to state the main theorem of this section.

Theorem 5.4. Let and be von Neumann algebras such that dec( ) is a non-measurable M N M⊗N cardinal, and suppose is properly infinite. If Φ ( ) is such that N ∈ CB M (Φ id ) (id Ψ) = (id Ψ) (Φ id ) ⊗ N M⊗ M⊗ ⊗ N for all Ψ ( ), then Φ must be normal. ∈ CB N Remark 5.5. The assumption that the decomposability number of be non-measurable, is very M⊗N weak and in fact just excludes a set-theoretic pathology. We briefly recall that a cardinal κ is said to be (real-valued) measurable if for every set Γ of cardinality κ, there exists a diffused probability measure on the power set P(Γ). – Measurability is a property of “large” cardinals ( 0 is of course ℵ not measurable). In order to demonstrate how weak the restriction to non-measurable cardinals is, let us just mention the fact (cf. [Kan–Mag 78], 1, p. 106, 108) that the existence of measurable § cardinals cannot be proved in ZFC (= the axioms of Zermelo–Fraenkel + the axiom of choice), and that it is consistent with ZFC to assume the non-existence of measurable cardinals ([Gar–Pfe 84], 4, Thm. 4.14, p. 972). For a further discussion, we refer to [Neu 01], and the references therein. § We begin now with the preparations needed for the proof of Theorem 5.4. We first note that, even though requiring a non-measurable decomposability number means only excluding exotic von Neumann algebras, it is already enough to ensure a very pleasant property of the predual. The latter is actually of great (technical) importance in the proof of Theorem 5.4 and is therefore stated explicitly in the following.

Proposition 5.6. Let be a von Neumann algebra. Then dec( ) is non-measurable if and only R R if the predual has Mazur’s property (i.e., w∗-sequentially continuous functionals on are w∗- R∗ R continuous – and hence belong to ). R∗ Proof. This is Thm. 3.11 in [Neu 01].

The crucial idea in the proof of Theorem 5.4 is to use a variant of the concept of a countable spiral nebula, which has been introduced in [Hof–Wit 97], 1.1 (cf. ibid., Lemma 1.5). §

Deﬁnition 5.7. Let be a von Neumann algebra. A sequence (αn, en)n N of ∗-automorphisms αn R ∈ on and projections en in will be called a countable reduced spiral nebula on if the following R R R holds true:

en en+1, WOT lim en = 1, αm(em) αn(en) for all m, n N0, m = n. ≤ − n ⊥ ∈ 6 The following Proposition is the crucial step in order to establish Theorem 5.4. It shows that a countable reduced spiral nebula may be used to derive a result on automatic normality – and not only on automatic boundedness, as is done in [Hof–Wit 97], Lemma 1.5.

Proposition 5.8. Let be a von Neumann algebra such that dec( ) is non-measurable, and suppose R R that there is a countable reduced spiral nebula (αn, en) on . R

18 k(k+1) N (i) Put lk 2 , and let F be a free ultraﬁlter on . Then the operators p p 1 Ψn w∗ lim α− − k F lk+n p → are completely positive and unital.

w 1 ∗ (ii) If Φ ( ) commutes with all elements of the set S αn− n N0 ( ), then Φ ∈ B R { | ∈ } ⊆ CP R is normal. p Proof. The statement (i) is clear. In order to prove (ii), in view of Proposition 5.6, we only have to show that if (xn)n N0 Ball( ) converges w∗ to 0, then the same is true for the sequence (Φ(xn))n. ∈ ⊆ R It suﬃces to prove that if (Φ(xni ))i is a w∗-convergent subnet, then its limit must be 0. The projections αn(en) being pairwise orthogonal, noting that lκ+1 = lκ + κ + 1, we see that

κ ∞ x αl +ν(el +νxνel +ν) . X X κ κ κ ∈ R e p κ=0 ν=0 Now we deduce (cf. the proof of [Hof–Wit 97], Lemma 1.5) that

Ψn(x) = xn for all n N0. (13) ∈ e Let Ψ w∗ limj Ψn Ball( ( )) be a w∗-cluster point of the net (Ψn )i. Then Ψ S, and − ij ∈ CB R i ∈ we obtain,p using (13):

lim Φ(xn ) = lim Φ(Ψn (x)) = lim Ψn (Φ(x)) i i i i i i e e = lim Ψn (Φ(x)) = Ψ(Φ(x)) j ij e e = Φ(Ψ(x)) = Φ(lim Ψn (x)) j ij e e = Φ(lim xn ) = Φ(0) = 0, j ij which ﬁnishes the proof.

We are now suﬃciently prepared for Theorem 5.4.

Proof. Since is properly infinite, it is isomorphic to (`2(Z)); cf., e.g., Appendix C, “The- N N ⊗B orem”, in [Van 78]. By applying the argument in the first part of the proof of Prop. 2.2 in [Hof–Wit 97], we can construct a countable reduced spiral nebula (αn, en) on (`2(Z)). Hence, B (αn, en) (id id αn, 1 1 en) is a countable reduced spiral nebula on (`2(Z)) = M⊗ N ⊗ M⊗ N ⊗ M⊗N ⊗B e e .p We will now apply Proposition 5.8 to the von Neumann algebra and the mapping M⊗N M⊗N Φ id ( ) to show that the latter, and hence Φ itself, is normal. – To this end, we only ⊗ N ∈ CB M⊗N w 1 ∗ have to prove that Φ id commutes with all elements of the set S αn− n N0 . But the ⊗ N { | ∈ } amplification in the first variable, χ , is w∗-w∗-continuous (this is establishedp e in the same fashion M as we did for χ in Theorem 3.1), and this entails that every element of S is of the form id Ψ M M ⊗ for some Ψ ( ). ∈ CB N

19 Remark 5.9. The assumption of being properly infinite may be interpreted as a certain richness N condition which is necessary to be imposed on the second von Neumann algebra. Indeed, if in contrast is, e.g., a finite type I factor, the statement is wrong in general. This is easily seen as N n n n follows: In this case, = (` ) = Mn for some n N, and hence ( ) = ( (` ) (` )) = N B 2 ∈ CB N T 2 ⊗B 2 ∗ ( (`n) (`n)) = σ( ). Now let be an arbitrary von Neumann algebra such that theb singular T 2 ⊗K 2 ∗ CB N M part ofb the Tomiyama–Takesaki decomposition of ( ) is non-trivial, i.e., s( ) = (0). Then CB M CB M 6 by Theorem 5.1, every Φ s( ) will satisfy the commutation relation in Theorem 5.4 for all ∈ CB M Ψ ( ) = σ( ), even though it is singular. ∈ CB N CB N We finish our discussion with considering again an arbitrary admissible pair ( , ). In view of M N Theorem 5.1, it would be natural, following the model of the tensor product of bounded operators on Hilbert spaces, to define a tensor product of Φ ( ) and Ψ σ( ) by setting: ∈ CB M ∈ CB N Φ Ψ (Φ id ) (id Ψ) = (id Ψ) (Φ id ), ⊗ ⊗ N M⊗ M⊗ ⊗ N p which defines an operator in ( ). It would be interesting to go even further and consider, CB M⊗N for arbitrary Φ ( ) and Ψ ( ), the two “tensor products” Φ 1Ψ and Φ 2Ψ, defined ∈ CB M ∈ CB N ⊗ ⊗ respectively through e e Φ 1Ψ (Φ id ) (id Ψ) ( ) ⊗ ⊗ N M⊗ ∈ CB M⊗N p and e p Φ 2Ψ (id Ψ) (Φ id ) ( ). ⊗ M⊗ ⊗ N ∈ CB M⊗N p By Theorem 5.4, we know thate thesep are different in general, even in the case of von Neumann algebras and . We wish to close this section by stressing the resemblance of these considerations M N to the well-known construction of the two Arens products on the bidual of a Banach algebra, which in general are different. From this point of view, the subalgebra σ( ) could play the role of what CB N in the context of Banach algebras is known as the topological centre (for the latter term, see, e.g., [Dal 00], Def. 2.6.19).

6 Uniqueness of the ampliﬁcation and a generalization of the Ge– Kadison Lemma

In the following, we will show that under different natural conditions, an algebraic amplification is uniquely determined. This leads in particular to a remarkable generalization of the so-called Ge–Kadison Lemma (see Proposition 6.2), which is obtained as an application of our Theorem 5.1. But before establishing positive results, we briefly point out why in general, an algebraic amplification is highly non-unique. In [Tom 70], p. 28, Tomiyama states, in the context of von Neumann algebras, that the “product projection” of two projections of norm one “might not be unique” (of course, in our case, the second projection of norm one would be the identity mapping). The following simple argument shows that whenever and are infinite-dimensional dual operator spaces, M N there are uncountably many different algebraic amplifications of a map Φ ( ) to a completely ∈ CB M bounded map on , regardless of Φ being normal or not. Namely, for any non-zero functional M⊗N ϕ ( ) which vanishes on ∨ , and any non-zero vector v , ∈ M⊗N ∗ M⊗N ∈ M⊗N χϕ,v(Φ) χ (Φ) ϕ, χ (Φ)( ) v N N − h N · i p 20 defines such an algebraic amplification. Furthermore, the amplification χϕ,v can of course be taken arbitrarily close to an isometry: for every ε > 0, an obvious choice of ϕ Nand v yields an algebraic amplification χϕ,v : ( ) ( ) such that N CB M −→ CB M⊗N ϕ,v (1 ε) Φ cb χ (Φ) cb (1 + ε) Φ cb − k k ≤ k N k ≤ k k for all Φ ( ). ∈ CB M We begin our investigation by noting a criterion which establishes the uniqueness of amplification for a large class of von Neumann algebras, though only assuming part of the requirements listed in Definition 2.2. Proposition 6.1. Let and be von Neumann algebras, where is an injective factor. If M N M χ : ( ) ( ) is a bounded linear mapping which satisfies the algebraic amplification 0 CB M −→ CB M⊗N condition (AAC) and properties (iii) and (iv) in Definition 2.2, then χ0 = χ . N Proof. Owing to [Cha–Smi 93], Thm. 4.2, we have the density relation:

w σ( ) ∗ = ( ). CB M CB M

Hence, due to their w∗-w∗-continuity (condition (iii)), χ and χ0 will be identical if they only σ σ N coincide on ( ). But if Φ ( ), by condition (iv), the mappings χ (Φ) and χ0(Φ) are CB M ∈ CB M N normal, so that to establish their equality, it suffices to show that they coincide on all elementary tensors S T . But this is true since they both fulfill condition (AAC). ⊗ ∈ M⊗N We will now present our generalization of the Ge–Kadison Lemma which was first obtained in [Ge–Kad 96] (ibid., Lemma F) and used in the solution of the splitting problem for tensor products of von Neumann algebras (in the factor case). Furtheron, a version of the Lemma was again used by Str˘atil˘aand Zsidóin [Str–Zsi 99] in order to derive an extremely general commutation theorem which extends both Tomita’s classical commutant theorem and the above mentioned splitting result ([Str–Zsi 99], Thm. 4.7). The result we give below generalizes the Lemma in the version as stated (and proved) in [Str–Zsi 99], 3.4, which the authors refer to as a “smart technical device”. In § [Str–Zsi 99], it is applied to normal conditional expectations of von Neumann algebras (ibid., 3.5); § of course, these are in particular completely positive. Our generalization yields an analogous result which in turn can be applied to arbitrary completely bounded mappings. This fact may be of interest in the further development of the subject treated in [Ge–Kad 96] and [Str–Zsi 99]. Furthermore, we prove that the Lemma not only holds for von Neumann algebras but for arbitrary admissible pairs, a situation which is not considered in the latter articles. Hence, our version is likely to prove useful in the interesting problem of obtaining a splitting theorem in the more general context of w∗-spatial tensor products of ultraweakly closed subspaces – a question which is actually hinted at in [Ge–Kad 96] (p. 457). Proposition 6.2. Let ( , ) be an admissible pair, and Φ ( ) an arbitrary completely M N ∈ CB M bounded operator. Suppose Θ: is any map which satisfies, for some non-zero M⊗N −→ M⊗N element n : ∈ N (i) Θ commutes with the slice maps id τn (τ ) M ⊗ ∈ N∗ (ii) Θ coincides with Φ id on n. ⊗ N M ⊗ 21 Then we must have Θ = Φ id . ⊗ N In particular, if Φ ( ) and (Φα)α ( ) is a bounded net converging w∗ to Φ, then w∗ ∈ CB M ⊆ CB M Φα id Φ id . ⊗ N −→ ⊗ N Remark 6.3. (i) The known version of the Lemma supposes that and are von Neumann M N algebras, and that the mapping Φ is normal and completely positive. As shown by the above, all these assumptions can be either weakened or even just dropped. (ii) The last statement in the Proposition is even true without assuming the net (Φα)α to be bounded, and in this generality follows directly from our Theorem 3.1 (property (iii) in Definition 2.2). Nevertheless, this “unbounded” version does not appear in the literature. (We remark that the assumption of boundedness is missing in the statement of Lemma 3.4 in [Str–Zsi 99], though needed in the proof presented there.) We shall point out below how the “bounded” version can be easily obtained by using the first part of the Proposition (cf. [Str–Zsi 99], 3.4). § Proof. Using the first result of the last section, our argument follows the same lines as the one given in [Str–Zsi 99]; we present the proof because of its brevity. – Fix a non-zero element m . We ∈ M obtain for every u , ρ and τ : ∈ M⊗N ∈ M∗ ∈ N∗ Θ(u), ρ τ m n = [(ρm id ) (id τn) Θ] (u) h ⊗ i ⊗ ⊗ N M ⊗ = [(ρm id ) Θ (id τn)] (u) by (i) ⊗ N M ⊗ = [(ρm id ) (Φ id ) (id τn)] (u) by (ii) ⊗ N ⊗ N M ⊗ = [(ρm id ) (id τn) (Φ id )] (u) by Theorem 5.1 ⊗ N M ⊗ ⊗ N = (Φ id )(u), ρ τ m n, h ⊗ N ⊗ i ⊗ which proves the first statement in the Proposition. In order to prove the second assertion, we only have to show that if Θ is any w∗-cluster point of the net (Φα id )α, then Θ = Φ id . ⊗ N ⊗ N But Θ trivially satisfies condition (ii) in our Proposition, and using the normality of the mappings id τn, another application of Theorem 5.1 shows that Θ also meets condition (i), whence the M ⊗ first part of the Proposition finishes the proof.

7 Completely bounded module homomorphisms and Tomiyama’s slice maps

The main objective of [May–Neu–Wit 89] was to establish the following remarkable property of the ampliﬁcation mapping

( ) ( ( )) Φ Φ ∞ ( ( ) ( )). CB B H 3 7→ ∈ CB B H ⊗B K If ( ) is a von Neumann subalgebra, and if Φ ( ( )) is an -bimodule homomor- R ⊆ B H ( ) ∈ CBR B H ( ) R phism, then Φ ∞ is an ( )-bimodule homomorphism, i.e., Φ ∞ ( )( ( ) ( )); R⊗B K ∈ CBR⊗B K B H ⊗B K see [May–Neu–Wit 89], Prop. 2.2 (and [Eff–Rua 88], Thm. 4.2, for a generalization). An important special situation, which nevertheless shows the strength and encodes the essential statement of the result, is the case where = C 1 ( ). Then the above theorem reads as follows: R B H ( ) Every Φ ( ( )) can be amplified to an 1 ( ) ( )-bimodule homomorphism ∗∗ ∈ CB B H B H ⊗ B K Φ( ) ( ( ) ( )). ∞ ∈ CB B H ⊗B K 22 The aim of this section is to prove an analogous result where now ( ) and ( ) are replaced B H B K by arbitrary von Neumann algebras and , using our amplification χ . In this section, and M N N M will always denote von Neumann algebras. N The proofs of [May–Neu–Wit 89], Prop. 2.2 and [Eff–Rua 88], Thm. 4.2 use either the Witt- stock decomposition theorem or a corresponding result by Paulsen ([Pau 86], Lemma 7.1) to reduce to the case of a completely positive mapping, where the assertion is then obtained by a fairly subtle analysis involving the Schwarz inequality. In contrast to this, the explicit form of χ provides us N with a direct route to the above announced goal; indeed, as we shall see, the statement asserting the module homomorphism property of the amplified mapping reduces to a corresponding statement about Tomiyama’s slice maps which in turn can be verified in an elementary fashion (see Lemma 7.1 below). In order to prove our result, a little preparation is needed. In the sequel, we restrict ourselves to left slice maps; analogous statements hold of course in the case of right slice maps. – As is well- known (and easy to see), Tomiyama’s left slice maps satisfy the following module homomorphism property: Lτ ((a 1 )u(b 1 )) = aLτ (u)b, ⊗ N ⊗ N where u , a, b , τ . In the following (elementary) lemma, we shall establish an ∈ M⊗N ∈ M ∈ N∗ analogous property involving elementary tensors of the form 1 a and 1 b for a, b . To M ⊗ M ⊗ ∈ N this end, we briefly recall that for every von Neumann algebra , the predual is an -bimodule N N∗ N in a very natural manner, the actions being given by

τ a, b = τ, ab and a τ, b = τ, ba , h · i h i h · i h i for a, b , τ . For later purposes, we note the (easily verified) equation: ∈ N ∈ N∗ (1 a) (ρ τ) (1 b) = ρ (a τ b), (14) M ⊗ · ⊗ · M ⊗ ⊗ · · where a, b , ρ , τ . We now come to ∈ N ∈ M∗ ∈ N∗ Lemma 7.1. Let and be von Neumann algebras. Then Tomiyama’s left slice map satisfies: M N Lτ ((1 a)u(1 b)) = Lb τ a(u), M ⊗ M ⊗ · · where u , a, b , τ . ∈ M⊗N ∈ N ∈ N∗ Proof. Due to the normality of left slice maps, it suffices to prove the statement for elementary tensors u = S T . But in this case, we obtain: ⊗ ∈ M⊗N Lτ ((1 a)u(1 b)) = Lτ (S aT b) M ⊗ M ⊗ ⊗ = τ, aT b S h i = b τ a, T S h · · i = Lb τ a(u), · · whence the desired equality follows.

The above statement will now be used to derive a corresponding module homomorphism property shared by the ampliﬁcation of a completely bounded mapping, as announced at the beginning of this section.

23 Theorem 7.2. Let and be von Neumann algebras. Then for every Φ ( ), the ampliﬁed M N ∈ CB M mapping χ (Φ) ( ) is an 1 -bimodule homomorphism. N ∈ CB M⊗N M ⊗ N Proof. As in the proof of Theorem 3.1, we denote by κ the pre-adjoint mapping of χ . Fix u N ∈ , a, b , ρ and τ . We write a0 1 a, b0 1 b. M⊗N ∈ N ∈ M∗ ∈ N∗ M ⊗ M ⊗ We ﬁrst note that p p

κ(ρ τ (a0ub0)) = κ(ρ (b τ a) u). (15) ⊗ ⊗ ⊗ · · ⊗ For by equation (6), in order to establish (15), we only have to show that

Lτ (a0ub0) = Lb τ a(u), · · which in turn is precisely the statement of Lemma 7.1. Now we obtain for Φ ( ): ∈ CB M χ (Φ)(a0ub0), ρ τ = Φ, κ(ρ τ (a0ub0)) h N ⊗ i h ⊗ ⊗ i (15) = Φ, κ(ρ (b τ a) u) h ⊗ · · ⊗ i (14) = Φ, κ[(b0 (ρ τ) a0) u] h · ⊗ · ⊗ i = χ (Φ)(u), b0 (ρ τ) a0 h N · ⊗ · i = a0χ (Φ)(u)b0, ρ τ , h N ⊗ i which ﬁnishes the proof.

We ﬁnish by remarking that Theorem 7.2 yields in particular the statement ( ) given above; ( ) ∗∗ for if Φ ( ( )), we have χ ( )(Φ) = Φ ∞ , as noted in section 4, equation (7). ∈ CB B H B K

Acknowledgments The author is indebted to Prof. Dr. G. Wittstock for many very fruitful discussions on the subject of the present paper.

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Author’s address: Department of Mathematical Sciences University of Alberta Edmonton, Alberta Canada T6G 2G1 E-mail: [email protected]