SELFDUAL HILBERT RIGHT W *-MODULES AND THEIR W *-TENSOR PRODUCTS

AND THEIR W *-TENSOR PRODUCTS

CORNELIU CONSTANTINESCU

We present some properties of selfdual Hilbert rightW^∗-modules and then discuss their exterior and interiorW^∗-tensor products, inclusive the KSGNS construction.

AMS 2000 Subject Classification: 46L06, 46L08.

Key words: selfdual Hilbert rightW^∗-modules,W^∗-tensor products, KS-GNS con- struction.

INTRODUCTION

Compared with the classical Hilbert spaces, the Hilbert rightC^∗-modu- les have very weak properties. Among these last, the selfdual Hilbert right W^∗-modules have better properties and come nearer to the Hilbert spaces.

We present some properties of the selfdual Hilbert right W^∗-modules in the first section after which we concentrate on their W^∗-tensor products.

Let E, F be W^∗-algebras and H (resp. K) a selfdual Hilbert right E- module (resp. F-module). We give a new proof of [C2], Theorem 2.4 that L_E(H) ¯⊗L_F(K) is isomorphic to L_E⊗F¯ (H⊗K) (Theorem 2.4). Let¯ ρ : E → L_F(K) be aW^∗-continuous completely positive linear map andρ∗ :L_E(H)→ L_F(H⊗_ρK) the associated unitalC^∗-homomorphism ([L], pages 42 and 48).

One defines the interiorW^∗-tensor product ofHandKasH⊗¯_ρK :=H\⊗_ρK.

We extend in a unique way ρ∗ to a unital W^∗-homomorphism ¯ρ∗ :L_E(H) → L_F(H⊗¯_ρK) (Theorem 3.1). A uniqueness of the corresponding W^∗-version of the KSGNS-construction ([L], page 52) is also proved (Proposition 4.1).

Finally, we give an example which points out the difference between ρ∗ and

¯

ρ∗ (Proposition 3.4). The proofs for all these results work simultaneously for the real and the complex case.

In general we use the notation and terminology of [C1] (see also [S-Z1]).

ForW^∗-tensor products ofW^∗-algebras we use [T] (see also [S-Z2]), for tensor products of Hilbert rightC^∗-modules we use [L], and for the exteriorW^∗-tensor products of selfdual Hilbert right W^∗-modules we use [C2].

In the sequel we give a list of some notation used in this paper.

REV. ROUMAINE MATH. PURES APPL.,55(2010),3, 159–196

1. Kdenotes the field of real or the field of complex numbers. The whole theory is developed in parallel for the real and complex case (but the proofs coincide). Ndenotes the set of natural numbers (06∈N) and for every n∈N we put Nn:=

m∈N|m≤n .R+ denotes the set of positive real numbers (0∈R+).

2. IfXis a set, then we denote for every subset AofX bye_A:=e^X_A the characteristic function ofA inX, i.e., the function onX equal to 1 onA and equal to 0 on X\A.

3. For every setA we denote by CardA its associated cardinal number and by P_f(A) the set of finite subsets of A.

4. IfE, F are vector spaces in duality then E_F denotes the vector space E endowed with the locally convex topology of pointwise convergence on F, i.e., with the weak topology σ(E, F).

5. IfE is a Banach space then E^# denotes its unit ballE^#:={x∈E | kxk ≤ 1}. If E has a unique predual (up to isomorphisms), then we denote by ¨E this predual and so byE_E¨ the vector spaceE endowed with the locally convex topology of pointwise convergence on ¨E.

6. If E, F are Banach spaces, then we denote by L(E, F) the Banach space of operators from E to F and for every ϕ ∈ L(E, F), ϕ⁰ denotes the transpose of ϕ. If in addition E and F have distinguished preduals ¨E and F¨, respectively, and if the map ϕ:E_E¨ →F_F¨ is continuous (or equivalently ϕ⁰( ¨F)⊂E), then we say that¨ ϕis W^∗-continuous and denote by ¨ϕ: ¨F → E,¨ b7→ϕ⁰bits pretranspose.

7. For alli, j we denote byδ_ij the Kronecker’s symbol δij :=

1 ifi=j 0 ifi6=j .

8. If E is a C^∗-algebra then we denote by PrE the set of orthogonal projections of E. If in addition E is unital then we denote by 1E its unit.

9. LetEbe aW^∗-algebra. For everyn∈N,E_n,ndenotes theW^∗-algebra ofn×nmatrices with entries inE. We use also the notationE_J,J in an obvious way for every finite set J. A C^∗-subalgebra F of E is called W^∗-subalgebra if it is a W^∗-algebra and if the inclusion map F_F¨ → E_E¨ is continuous. We say that a C^∗-subalgebra F of E generates E as a W^∗-subalgebra if every W^∗-subalgebra ofEcontainingF is equal toE; by [C1], Corollary 4.4.4.12 a), this is equivalent to the assertion that F is dense in EE¨ and by Kaplanski’s theorem ([C1], Corollary 6.3.8.7) to the assertion that F^# is dense in E^#_¨

E. Every C^∗-algebra generates its bidual as a W^∗-subalgebra. ¨E₊ denotes the convex cone of positive elements of the predual of E.

10. LetEbe aC^∗-algebra andH, K Hilbert rightE-modules. We denote by L_E(H, K) the Banach subspace of L(H, K) of adjointable operators, by

Lb_E(H, K) the Banach space of u∈ L(H, K) such that (ξ, x)∈H×E ⇒u(ξx) = (uξ)x,

by 1_H the identity mapH →H (which belongs toL_E(H) :=L_E(H, H)),and putHb :=Lb_E(H, E) and

h· |ξi:H→E, η7→ hη|ξi

for every ξ ∈H. If for every u∈Hb there is aξ ∈H withu :=h· |ξi then we say that H is selfdual. For (ξ, η)∈H×K we put

ηh· |ξi:H →K, ζ 7→ηhζ|ξi.

11. LetE be aW^∗-algebra and H, K Hilbert right E-modules. We put for a∈E¨ and (ξ, η)∈H×K,

(a, ξ) :] H →K, ζ 7→

hζ|ξi, a , (a, ξ, η) :^ L_E(H, K)→K, u7→

huξ|ηi, a

and denote by ¨H (by H) the closed vector subspace of^... H⁰ (of L_E(H, K)⁰) generated by

(a, ξ)] |(a, ξ)∈E¨×H (by

(a, ξ, η)^ |(a, ξ, η)∈E¨×H×K ).

IfH (resp. H andK) is selfdual, then ¨H (resp. H^...) is the predual ofH (resp.

of L_E(H, K))([C1], Proposition 5.6.3.3, [C1], Theorem 5.6.3.5 a)).

12. IfE is a W^∗-algebra and (H_i)i∈I is a family of Hilbert rightE-mo- dules, then we put

|

i∈I

H_i :=

ξ ∈Y

i∈I

H_i |the family (hξ_i|ξ_ii)_i∈Iis summable inE

,

W

|

i∈I

Hi:=

ξ ∈Y

i∈I

Hi |the family (hξ_i|ξii)i∈I is summable inEE¨

. |

i∈I

Hi is a Hilbert rightE-module and

W

|

E

is a selfdual Hilbert rightE-module ([C1], Proposition 5.6.4.1 c), [C1], Proposition 5.6.4.6 b), and [C1], Theorem 5.6.4.7 a))

13. denotes the algebraic tensor product of vector spaces.

14. IfE, F areW^∗-algebras and H (resp. K) is a selfdual Hilbert right E- (resp. F-) module, then we denote by H⊗K¯ theW^∗-tensor product of H and K, which is a selfdual Hilbert right E⊗F-module ([C2], Definition 2.3).¯ In contrast to the case of W^∗-algebras, the predual of H⊗K¯ is in general not isomorphic to the projective tensor product of ¨H and ¨K.

15. ≈means: isomorphic.

16. pa and T(Definition 1.1).

17. u⊗v¯ (Proposition 2.6).

18. ¯⊗_ρand ¯ρ∗ (Theorem 3.1).

19. F^c (Lemma 4.3).

20. Acts irreducibly (Definition 4.4).

1. SELFDUAL HILBERT RIGHT W^∗-MODULES

Definition 1.1. LetE be aW^∗-algebra andH a Hilbert rightE-module.

For every a∈ E¨₊ we put p_a :H → R+, ξ 7→

hξ|ξi, a¹₂

; it is a seminorm (by Schwarz’ inequality). We denote by T the locally convex topology onH generated by the family (pa)_a∈E¨+ of seminorms on H.

Lemma 1.2. Let E be a W^∗-algebra and H, K Hilbert right E-modules.

a)For (a, ξ)∈E¨+×H, pa(ξ)≤ kak¹²kξk.

b) For (a, ξ)∈E¨₊×H andu∈ L_E(H, K), p_a(uξ)≤ kukp_a(ξ).

c)For (a, ξ)∈E¨×H, k(a, ξ)k ≤ kak] ¹² p_|a|(ξ).

d) For(a, ξ, η)∈E¨×H×K anda=x|a| the polar representation of a, k(a, ξ, η)k ≤^ p_x|a|x^∗(ξ)p_|a|(η).

e)The relative topology T onH^# is finer than the topology of H^#_¨

H. f)For u∈ L_E(H), ξ, η∈H, anda∈E¨₊,

huξ|ηi, a

≤ kukp_a(ξ)pa(η), i.e., the sesquilinear map

H×H →E_E¨, (ξ, η)7→ huξ|ηi is continuous with respect to the topology T.

Proof. a) is obvious.

b) We have pa(uξ)²=

huξ|uξi, a

=

hu^∗uξ|ξi, a

≤ kuk²

hξ|ξi, a , p_a(uξ)≤ kukp_a(ξ).

c) Leta= x|a| be the polar representation of a. For ζ ∈H, by a) and Schwarz’ inequality,

ζ,(a, ξ)]

2 =

hζ|ξi, a

2 =

hζ|ξi, x|a|

2=

=

hζx|ξi,|a|

2 ≤

hζx|ζxi,|a| hξ|ξi,|a|

≤

≤ k|a|k kζxk²p_|a|(ξ)² ≤ kak kζk²p_|a|(ξ)²,

so

k(a, ξ)k ≤ kak] ¹²p_|a|(ξ).

d) Foru∈ L_E(H, K), by Schwarz’ inequality,

u, (a, ξ, η)^

2=

huξ|ηi, a

2 =

huξ|ηi, x|a|

2=

huξx|ηi,|a|

2 ≤

≤

huξx|uξxi,|a| hη|ηi,|a|

=

hu^∗uξx|ξxi,|a|

p_|a|(η)² ≤

≤ kuk²

hξx|ξxi,|a|

p_|a|(η)² =kuk²

x^∗hξ|ξix,|a|

p_|a|(η)²=

=kuk²

hξ|ξi, x|a|x^∗

p_|a|(η)²=kuk² p_x|a|x^∗(ξ)² p_|a|(η)², so

k(a, ξ, η)k ≤^ p_x|a|x^∗(ξ)p_|a|(η).

e) SinceH^# is equicontinuous, H^#_¨

H is the topology on H^# of pointwise convergence on the set of linear combinations of elements from

(a, ξ)] |(a, ξ)∈E×H and by c), Tis finer than this topology.

f) By d) (and [C1], Theorem 5.6.3.5 b)),

huξ|ηi, a =

u, (a, ξ, η)^

≤ kuk k(a, ξ, η)k ≤ kukp^ _a(ξ)pa(η).

Lemma1.3. Let E be aW^∗-algebra, K a Hilbert rightE-module, andH a right E-submodule of K.

a) If H is dense in K_K¨, then K_H¨ (with obvious notation) is Hausdorff.

b) If in addition K is selfdual, then K^#_¨

K =K^#_¨

H is compact.

Proof. a) Letη ∈K with

η, (a, ξ)]

= 0

for all (a, ξ)∈E¨×H. LetFbe a filter on H converging toη inK_K¨. Then hη|ηi, a

=

η, (a, η)]

= lim

ξ,F

ξ, (a, η)]

= lim

ξ,F

hξ|ηi, a

=

= lim

ξ,F

hη|ξi, a^∗

= lim

ξ,F

η, (a^^∗, ξ)

= 0.

Thus

hη|ηi, a

= 0, hη|ηi= 0, η= 0.

b) Since K is selfdual, by [C1], Proposition 5.6.3.3 a) ⇒ b), K^#_¨

K is compact, so by a), K^#_¨

K =K^#_¨

H.

Proposition 1.4. Let E be a C^∗-subalgebra of a W^∗-algebra F, H, K selfdual Hilbert right F-modules, andL (resp. M) a closed right E-submodule of H (resp. of K) such that L^# is dense in H^#_¨

H (resp. M^# is dense inK^#_¨

K) and such that the range of the scalar products of L and M lie in E.

a) For every u ∈ L_E(L, M) there is a unique u¯ ∈ L_F(H, K) extending u. Moreover u¯^∗ =u^∗ and u¯ is the unique extension of u with u¯:HH¨ →KK¨

continuous.

b)

¯

u|u∈ L_E(L, M) =

v∈ L_F(H, K)|v(L)⊂M .

c) If there is a (ξ, η) ∈ L×M with u = ηh· |ξi then u¯ = ηh· |ξi ∈ K_E(H, K).

d) If u ∈ L_E(L, M) is an isomorphism then u¯ is an isomorphism of Hilbert right F-modules.

e) If H =K and L=M, then the map, L_E(L)→ L_F(H) u7→ u¯ is an injective C^∗-homomorphism.

Proof. a) First we remark that by our hypothesesL and M are Hilbert right E-modules so thatL_E(L, M) is well-defined. SinceLis dense inH_H¨ the uniqueness follows from the fact that every operator ofL_F(H, K) is continuous as a map HH¨ →KK¨ ([C1], Proposition 5.6.3.4 c)).

Letξ0 ∈H^#, η ∈ M, and a∈F¨, and let F(ξ0) be the induced filter on L^# by the neighbourhood filter of ξ0 in H^#_¨

H. Without loss of generality we may assumekuk ≤1. Then

hξ₀|u^∗ηi, a

=

ξ0,(a, u^^∗η)

= lim

ξ,F(ξ0)

ξ, (a, u^^∗η)

=

= lim

ξ,F(ξ0)

hξ|u^∗ηi, a

= lim

ξ,F(ξ0)

huξ|ηi, a

= lim

ξ,F(ξ0)

uξ,(a, η)] . By Lemma 1.3 b), K^#_¨

K =K^#_¨

M is compact. Thus lim_ξ,F(ξ0)uξexists inK^#_¨

K and if we put u₀ :H^#→K^#, ξ₀7→lim_ξ,F(ξ0)uξ, then

hξ₀|u^∗ηi, a

=

u0ξ0,(a, η)]

=

hu₀ξ0|ηi, a , hξ₀|u^∗ηi=hu₀ξ0|ηi.

Similarly, there is a map (u^∗)₀ :K^#→H^#with hη₀|uξi=h(u^∗)0η0|ξi for all (ξ, η0)∈L×K^#.

For (a, ξ₀, η₀)∈F¨×H^#×K^#, hξ₀|(u^∗)₀η₀i, a

=

ξ₀,z ^}| { (a,(u^∗)₀ η₀)

= lim

ξ,F(ξ0)

ξ, z ^}| { (a,(u^∗)₀ η₀)

=

= lim

ξ,F(ξ0)

hξ|(u^∗)0η0i, a

= lim

ξ,F(ξ0)

huξ|η0i, a

=

= lim

ξ,F(ξ0)

uξ,(a, η^₀)

=

u₀ξ₀,(a, η^₀)

=

hu₀ξ₀|η₀i, a , hξ₀|(u^∗)₀ η₀i=hu₀ξ₀|η₀i.

Thus, if we denote by ¯u the extension of u0 to a map H→K, then

¯

u∈ L_F(H, K) with ¯u^∗=u^∗.

b), c), d), and e) follow from the uniqueness in a) and (for d)) from the last formula in the proof of a).

Corollary1.5. LetE be aC^∗-subalgebra of aW^∗-algebraF generating it as a W^∗-subalgebra, K a selfdual Hilbert right F-module, and H a closed right E-submodule ofK such that the range of its scalar product lies in E.

a)The following are equivalent:

a1)K is the extension ofH to a selfdual Hilbert rightF-module([C2], Proposition 1.3 f)).

a₂) H^# is dense in K^#_¨

K.

b) If the conditions ofa)are fulfilled, then the vector subspace ofL_F(K) generated by

ηh· |ξi |ξ, η∈H is dense in L_F(K)^...

K. c)If E =F then the following are equivalent:

c1) Hb is isomorphic to K.

c2) H^# is dense in K^# with respect to the topology T.

c3) H^# is dense in K^#_¨

K.

Proof. a₁)⇒a₂) follows from [C2], Lemma 1.2 a) and [C2], Proposition 1.3 f1), f2).

a₂)⇒a₁). IfLdenotes the extension ofH to a selfdual Hilbert rightF- module, then by a₁)⇒a₂) and Proposition 1.4 d) there is an isomorphismL→ K of Hilbert rightF-modules the restriction toHof which is the identity map.

b) follows from [C2], Proposition 1.3 f₄).

c) By [C2], Proposition 1.3 c), Hb is equal to the extension of H to a selfdual Hilbert right F-module, so c₁)⇔c₃) follows from a).

c1)⇒c2) follows from the proof of [C1], Proposition 5.6.2.9 d).

c₂) ⇒ c₃) follows from the fact that the topology induced by T on K^# is finer than the topology of K^#_¨

K (Lemma 1.2 e)).

Lemma 1.6. Let E be a W^∗-algebra, (Hi)i∈I a finite family of Hilbert right E-modules, H := |

i∈I

H_i, for every i∈I let ψ_i :H_i→H be the natural inclusion, and for everyu∈ |

i∈I

Hb_i puteu:H→E,ξ7→P

i∈Iu_iξ_i.Thenue∈Hb

for every u ∈ |

i∈I

Hbi and the map ϕ : |

i∈I

Hbi → H,b u 7→ eu is an isomorphism of Hilbert right E-modules with inverse mapψ:Hb → |

i∈I

Hb_i,v 7→(v◦ψ_i)i∈I. Proof. For (ξ, x)∈H×E,

eu(ξx) =X

i∈I

ui(ξix) =X

i∈I

(uiξi)x= (uξ)x,e so eu∈H. Moreover ([C1], Theorem 5.6.2.11),b

(eux)ξ=x^∗(uξ) =e x^∗X

i∈I

u_iξ_i =X

i∈I

(u_ix)ξ_i =uxξ,f so uxf =ux. Thus,e

ϕ∈Lb_E |

i∈I

Hbi,Hb

=Lb_E |

i∈I

Hbi,Hb

([C1], Proposition 5.6.2.4, [C1], Theorem 5.6.4.7 a)). It is easy to see that ψ ∈ Lb_E

H,b |

i∈I

Hb_i

and that ψ is the inverse of ϕ. Thus, ϕ is bijective. In order to show that ϕis an isomorphism it is therefore sufficient to show that ψ preserves the scalar products.

We identify H_i with a right Hilbert E-submdule of Hb_i for every i ∈ I and similarly for H and Hb using the mapξ 7→ h· |ξi ([C1], Theorem 5.6.2.11 e)). For ξ, η∈H,

ψh· |ξi= (h· |ξi ◦ψi)i∈I = (h· |ξii)_i∈I, hψh· |ξi |ψh· |ηi i=h(h· |ξ_ii)_i∈I|(h· |η_ii)_i∈Ii=

=X

i∈I

hh· |ξii | h· |ηii i=X

i∈I

hξ_i|ηii=hξ|ηi. Letξ₀, η₀ ∈Hb and letFbe a filter onHconverging toξ₀inHb ^..

z}|{

Hb

(Corollary 1.5 c1)⇒c3). For (a, η)∈E¨×H, by the above and [C1], Proposition 5.6.3.4 c),

hξ₀|ηi, a

=

ξ₀,(a, η)]

= lim

ξ,F

ξ, (a, η)]

= lim

ξ,F

hξ|ηi, a

=

= lim

ξ,F

hψξ|ψηi, a

= lim

ξ,F

ψξ,(a, ψη)^

=

ψξ₀,(a, ψη)^

=

hψξ₀|ψηi, a and sohξ₀|ηi=hψξ₀|ψηi. It follows by [C1], Proposition 5.6.3.4 c) again that

hξ₀|η0i, a

=

ξ0,(a, η^0)

= lim

ξ,F

ξ, (a, η^0)

=

= lim

ξ,F

hξ|η0i, a

= lim

ξ,F

hψξ|ψη0i, a

= lim

ξ,F

ψξ,(a, ψη^0)

=

=

ψξ₀,(a, ψη^₀)

=

hψξ₀|ψη₀i, a , and so hξ₀|η₀i=hψξ₀|ψη₀i.

Lemma 1.7. Let E be a C^∗-subalgebra of a unital C^∗-algebra F, (H_i)i∈I

a family of Hilbert right E-modules, and for everyi∈I letKi be the extension of H_i to a Hilbert right F-module ([C2], Proposition 1.3 c)). Then |

i∈I

K_i is isomorphic to the extension of |

i∈I

H_i to a Hilbert rightF-module.

Proof. Put

H:= |

i∈I

Hi, K := |

i∈I

Ki, and consider the linear map

u:HF → |

i∈I

(H_iF), ξ⊗x7→(ξ_i⊗x)i∈I. For (ξ, x),(η, y)∈H×F,

hu(ξ⊗x)|u(η⊗y)i=h(ξ_i⊗x)i∈I|(η_i⊗y)i∈Ii=

=X

i∈I

hξ_i⊗x|η_i⊗yi=X

i∈I

y^∗hξ_i|η_iix=y^∗

X

i∈I

hξ_i|η_ii

x=

=y^∗hξ|ηix=hξ⊗x|η⊗yi.

Thus u is well-defined and factorizes to a map (with the notation of [C2], Proposition 1.3 c))

v: (HF)/(H•F)→K

preserving the scalar products, which extends by continuity to a mapH^F →K preserving the scalar products. Since this map is obviously surjective, it is an isomorphism of Hilbert right F-modules.

Proposition 1.8. Let E be a C^∗-subalgebra of a W^∗-algebra F genera- ting it as aW^∗-subalgebra,(Hi)i∈Ia family of Hilbert rightE-modules, and for every i∈I let K_i be the extension of H_i to a selfdual Hilbert right F-module.

Then

W

|

i∈I

K_i is isomorphic to the extension of |

i∈I

H_i to a selfdual Hilbert right F-module. If for every i∈I there is a p_i∈PrE with H_i =p_iE then

W

|

i∈I

K_i≈ ^W|

i∈I

(p_iF).

Proof. Put

H := |

i∈I

H_i, K :=

W

|

i∈I

K_i.

Let η∈K^#, (a, ζ)∈F¨×K, andε >0. By [C1], Proposition 5.6.4.6 c), there is a finite subset J ofI such that

ηeJ−η, (a, ζ)] < ε

2, where

(ηe_J)_i:=

ηi ifi∈J 0 ifi∈I\J . Since J is finite, by Lemma 1.6 and Lemma 1.7, |

i∈J

K_i is the extension of |

i∈J

Hi to a selfdual Hilbert rightF-module. Thus by Corollary 1.5 a1)⇒a2), there is ξ∈

|

i∈I

Hi

#

with

ξ−ηeJ,(a, ζ)] < ε

2. It follows

ξ−η, (a, ζ)] < ε.

Thus,

|

i∈I

Hi

#

is dense in K^#_¨

K. By Corollary 1.5, a2) ⇒ a1),

W

|

i∈I

Ki is isomorphic to the extension of |

i∈I

Hi to a selfdual Hilbert rightF-module.

If for everyi∈I there ispi ∈PrE withHi=piE, then by [C2], Propo- sition 1.3 e) (and [C1], Proposition 5.6.2.3), K_i is isomorphic top_iF and the assertion follows.

Proposition 1.9. Let E be a W^∗-algebra, H, K selfdual Hilbert right E-modules, andL (resp.M) a Hilbert rightE-submodule of H (resp.K) such that L^# is dense in H^#_¨

H (resp.M^# is dense in K^#_¨

K).

a)The vector subspace of H^... generated by (a, ξ, η)^ |(a, ξ, η)∈E¨×L×M is dense in H.^...

b) K_E(L, M) (considered in a natural way as a subset of K_E(H, K)) is dense in L_E(H, K)^...

H.

Proof. a) Let (a, ξ₀, η₀)∈E×H¨ ^#×K^#andε >0. Leta=x|a|be the po- lar representation ofa. By Corollary 1.5 c₃)⇒c₂), there is (ξ, η)∈L^#×M^# such that

p_x|a|x^∗(ξ−ξ0)< ε, p_|a|(η−η0)< ε.

Then

(a, ξ, η)^ −(a, ξ^0, η0) =z ^}| {

(a, ξ−ξ0, η−η0) +z ^}| {

(a, ξ−ξ0, η0) +z ^}| { (a, ξ0, η−η0)

so, by Lemma 1.2 d),

k(a, ξ, η)^ −(a, ξ^0, η0)k ≤

≤ kz ^}| {

(a, ξ−ξ0, η−η0)k+kz ^}| {

(a, ξ−ξ0, η0)k+kz ^}| { (a, ξ0, η−η0)k ≤

≤p_x|a|x^∗(ξ−ξ₀)p_|a|(η−η₀) +p_x|a|x^∗(ξ−ξ₀)p_|a|(η₀) +p_x|a|x^∗(ξ₀)p_|a|(η−η₀)<

< ε²+εp_|a|(η₀) +εp_x|a|x^∗(ξ₀).

Since εis arbitrary, (a, ξ^₀, η₀) belongs to the closure inH^... of the set (a, ξ, η)^ |(a, ξ, η)∈E¨×L×M .

The assertion follows.

b) Denote by F the closure ofK_E(L, M) in L_E(H, K)^...

H. Let (ξ₀, η₀)∈H×K, (a, ξ, η)∈E¨×H×K, and ε >0. There is ¯η∈M with

η¯−η0,z ^}| { (hξ|ξ0ia, η)

< ε 2 and ¯ξ ∈L with

ξ¯−ξ₀,z ^}| { (hη|η¯ia^∗, ξ)

< ε 2. Then

η¯

· ξ¯

−η₀h· |ξ₀i,(a, ξ, η)^ ≤

≤

η¯

· ξ¯−ξ0

,(a, ξ, η)^ +

(¯η−η0)h· |ξ0i,(a, ξ, η)^ =

=

h¯η|ηi ξ

ξ¯−ξ0

, a +

h¯η−η0|ηi hξ|ξ0i, a =

= ξ

ξ¯−ξ0

, ah¯η|ηi +

hη¯−η0|ηi,hξ|ξ0ia =

=| ξ¯−ξ0|ξ

,hη|η¯ia^∗

|+|

hη¯−η0|ηi,hξ|ξ0ia

|=

=

ξ¯−ξ₀,z ^}| { (hη|η¯ia^∗, ξ)

+

η¯−η₀,z ^}| { (hξ|ξ₀ia, η)

< ε 2 +ε

2 =ε.

Thus η₀h· |ξ₀i ∈ F and so K_E(H, K) ⊂F. By [C1], Proposition 5.6.5.13 e) (replacing there H by H| K), K_E(H, K) is dense in L_E(H, K)_H^... so F = L_E(H, K).

Lemma 1.10. Let E be a W^∗-algebra, (H_i)i∈I a family of Hilbert right E-modules, and H :=

W

|

i∈I

H_i. Then the vector subspace of H^... generated by (a, ξ, η)^ |(a, ξ, η)∈E¨×H×H,

i∈I |ξ_i 6= 0 or η_i 6= 0 is finite is dense in H.^...

Proof. Let (a, ξ, η) ∈ E¨ ×H ×H and ε > 0 and let a = x|a| be the polar representation of a. There is a finite subset J of I such that defining ξeJ, ηeJ ∈H by

(ξe_J)_i :=

ξi ifi∈J

0 ifi∈I\J , (ηe_J)_i :=

ηi ifi∈J 0 ifi∈I\J , we have

px|a|x^∗(ξ−ξeJ)< ε

2(p_|a|(η) + 1), p|a|(η−ηeJ)< ε

2(p_x|a|x^∗(ξ) + 1). It follows from

(a, ξ, η)^ −z ^}| {

(a, ξe_J, ηe_J) =z ^}| {

(a, ξ−ξe_J, η) +z ^}| { (a, ξe_J, η−ηe_J) by Lemma 1.2 d),

k(a, ξ, η)^ −z ^}| {

(a, ξe_J, ηe_J)k ≤ kz ^}| {

(a, ξ−ξe_J, η)k+kz ^}| { (a, ξe_J, η−ηe_J)k ≤

≤p_x|a|x^∗(ξ−ξeJ)p_|a|(η) +p_x|a|x^∗(ξeJ)p_|a|(η−ηeJ)<

< ε p_|a|(η)

2(p_|a|(η) + 1)+ ε p_x|a|x^∗(ξ)

2(p_x|a|x^∗(ξ) + 1) < ε 2 +ε

2 =ε.

The assertion follows.

Lemma 1.11. Let F be a W^∗-algebra, K a selfdual Hilbert right F- module, E a W^∗-subalgebra of L_F(K), (pj)j∈J a finite family in PrE, and for every x∈ L_E

|

j∈J

(pjE)

put

xe: |

j∈J

pj(K)→ |

j∈J

pj(K), ξ7→

X

k∈J

xjkξk

j∈J

. Then xe∈ L_F

|

j∈J

p_j(K)

for every x∈ L_E |

j∈J

(p_jE)

and the map L_E

|

j∈J

(pjE)

→ L_F |

j∈J

pj(K)

, x7→xe is an injective W^∗-homomorphism.

Proof. By [C1], Theorem 5.6.6.1 f), L_E

|

j∈J

(p_jE)

≈

x∈E_J,J |j, k∈J ⇒x_jk ∈p_jEp_k

so the above maps are well-defined. It is easy to see that the last map is an injective C^∗-homomorphism. By [C1], Theorem 5.6.6.5 c), it is a W^∗- homomorphism.

Theorem 1.12. Let F be a W^∗-algebra, K a selfdual Hilbert right F- module,E a W^∗-subalgebra ofL_F(K), and(p_i)i∈I a family in PrE. For every J ⊂I put

H_J :=

W

|

i∈J

(p_iE), H :=H_I, L_J :=

W

|

i∈J

p_i(K), L:=L_I, p_J :H→H_J, ξ 7→(ξ_i)i∈J, q_J :L→L_J, η7→(η_i)i∈J, and for J ∈P_f(I),

ϕJ :L_E(HJ)→ L_F(LJ), x7→ex the injective W^∗-homomorphism defined in Lemma 1.11 and

ψ_J :L_E(H)→ L_F(L), x7→q_J^∗ϕ_J(p_Jxp^∗_J)q_J.

a) If J₀ ∈ P_f(I) and ξ, η ∈ L with i ∈ I \J₀ ⇒ ξ_i = η_i = 0, then for every x∈ L_E(H) andJ0 ⊂J ∈P_f(I),

h(ψ_Jx)ξ|ηi=h(ψ_J0x)|ηi.

b)We denote byFthe upper section filter ofP_f(I). Then forx∈ L_E(H), lim_J,Fψ_Jx exists in L_F(L)^...

L; we put

θ:L_E(H)→ L_F(L), x7→lim

J,Fψ_Jx (in L_F(L)^...

L).

c) θ is injective, involutive, linear, continuous (with kθk ≤1), and W^∗- continuous.

d) If x, y∈ L_E(H) thenlim_J,Fxp^∗_JpJy=xy (in L_E(H)_H^...).

e)θ is an injective W^∗-homomorphism.

f)L_E(H) is isomorphic to a W^∗-subalgebra of L_F(L).

Proof. a) With the notation of Lemma 1.11,

h(ψ_Jx)ξ|ηi=hq_J^∗ϕ_J(p_Jxp^∗_J)q_Jξ|ηi=hϕ_J(p_Jxp^∗_J)q_Jξ|q_Jηi=

= X

i,j∈J

h(p_Jxp^∗_J)ijξj|ηii= X

i,j∈J₀

h(p_Jxp^∗_J)ijξj|ηii=h(ψ_J0x)ξ|ηi. b) Let (b, ξ, η)∈F¨×L×Lsuch that

J0 :=

i∈I |ξi 6= 0 orηi6= 0 ∈P_f(I).

For J₀ ⊂J ∈P_f(I), by a), ψJx, (b, ξ, η)^

=

h(ψ_Jx)ξ|ηi, b

=

h(ψ_J0x)ξ|ηi, b

=

ψJ0x, (b, ξ, η)^ so that

limJ,F

ψ_Jx,(b, ξ, η)^

=

ψ_J0x,(b, ξ, η)^ . By Lemma 1.10, lim_J,Fψ_Jx exists inL_F(L)^...

L.

c) Forx∈ L_E(H) andJ ∈P_f(I),

(ψ_Jx)^∗ =q^∗_Jϕ_J(p_Jx^∗p^∗_J)q_J =ψ_Jx^∗, kψ_Jxk ≤ kxk,

so θ is involutive and continuous with kθk ≤ 1. It is easy to see that θ is injective.

Let (b, ξ, η)∈F¨×L×Lsuch that J0 :=

i∈I |ξi 6= 0 orηi 6= 0 ∈P_f(I) and x∈ L_E(H). We put

ρ:L_E(H)→ L_E(HJ0), u7→pJ0up^∗_J0, τ :L_F(LJ0)→ L_F(L), v7→q^∗_J0vqJ0. By a) and b),

θx,(b, ξ, η)^

=

h(θx)ξ|ηi, b

=

h(ψ_J0x)ξ|ηi, b

=

= q^∗_J0ϕ_J0(p_J0xp^∗_J0)q_J0ξ|η , b

=

h(τ ϕ_J0ρx)ξ|ηi, b

=

τ ϕ_J0ρx, (b, ξ, η)^ . By Lemma 1.11, ϕJ0 isW^∗-continuous. This is obvious for τ and ρ so

θx,(b, ξ, η)^

=

x, ρ¨ϕ¨J0τ¨(b, ξ, η)^ , θ⁰(b, ξ, η) = ¨^ ρϕ¨_J0τ¨(b, ξ, η)^ ∈H .^...

By Lemma 1.10, θ⁰(L)^... ⊂H, i.e.,^... θ isW^∗-continuous.

d) Let (a, α, β)∈E¨×H×H and leta=z|a|be the polar representation of a. ForJ ∈P_f(I),

xy−xp^∗_Jp_Jy,(a, α, β)^

=

x(1_H −p^∗_Jp_J)y,(a, α, β)^

=

=

hx(1_H −p^∗_JpJ)yα|βi, a

=

hx(1_H−p^∗_JpJ)yαz|βi,|a|

=

=

hyαz|(1_H −p^∗_Jp_J)x^∗βi,|a|

so, by Schwarz’ inequality,

xy−xp^∗_Jp_Jy,(a, α, β)^

2 ≤

≤

hyαz|yαzi,|a| h(1_H −p^∗_Jp_J)x^∗β|(1_H −p^∗_Jp_J)x^∗βi,|a|

=

=

hyαz|yαzi,|a| h(1_H −p^∗_Jp_J)x^∗β|x^∗βi,|a|

=

=

hyαz|yαzi,|a| 1_H −p^∗_Jp_J,z ^}| { (|a|, x^∗β, x^∗β)

. It follows

limJ,F

xy−xp^∗_Jp_Jy, (a, α, β)^

= 0, limJ,Fxp^∗_JpJy=xy (inL_E(H)_H^...).

e) By c), we have only to prove thatθis multiplicative. Letx, y∈ L_E(H), (b, ξ, η)∈F¨×L×L with

J0:=

i∈I |ξi6= 0 orηi 6= 0 ∈P_f, and ε >0. By a), b), c), and d),

hθ(xy)ξ|ηi, b

=

θ(xy),(b, ξ, η)^

=

= lim

J,F

θ(xp^∗_Jp_Jy),(b, ξ, η)^

= lim

J,F

hψ_J0(xp^∗_Jp_Jy)ξ|ηi, b so there is a J₁∈P_f(I), J₀⊂J₁, such that

hψ_J0(xp^∗_Jp_Jy)ξ|ηi − hθ(xy)ξ|ηi, b < ε

3 for every J ∈P_f(I) withJ1 ⊂J. By a),

(1)

hψ_J(xp^∗_Jp_Jy)ξ|ηi − hθ(xy)ξ|ηi, b < ε

3 for every J ∈P_f(I) withJ₁ ⊂J.

There is aJ ∈P_f(I), J1⊂J, with

(2)

h(θx)(θy)ξ|ηi − h(θx)(ψ_Jy)ξ|ηi, b < ε

3 and a J⁰∈P_f(I), J ⊂J⁰, with

(3)

h(θx)(ψ_Jy)ξ|ηi − h(ψ_J⁰x)(ψ_Jy)ξ|ηi, b < ε

3. By a) and Lemma 1.11,

h(ψ_J⁰x)(ψJy)ξ|ηi=h(ψ_J⁰x)q_J^∗ϕJ(pJyp^∗_J)qJξ|ηi=

=h(ψ_Jx)q_J^∗ϕ_J(p_Jyp^∗_J)q_Jξ|ηi=hq_J^∗ϕ_J(p_Jxp^∗_J)q_Jq_J^∗ϕ_J(p_Jyp^∗_J)q_Jξ|ηi=

=hq^∗_Jϕ_J(p_Jxp^∗_J)ϕ_J(p_Jyp^∗_J)q_Jξ|ηi=hq^∗_Jϕ_J(p_Jxp^∗_Jp_Jyp^∗_J)q_Jξ|ηi=

=hψ_J(xp^∗_JpJy)ξ|ηi. so by (3),

(4)

h(θx)(ψ_Jy)ξ|ηi − h(ψ_J(xp^∗_Jp_Jy)ξ|ηi, b < ε

3. Thus by (1), (2), and (4),

(θx)(θy)−θ(xy),(b, ξ, η)^ =

h(θx)(θy)ξ|ηi − hθ(xy)ξ|ηi, b ≤

≤

h(θx)(θy)ξ|ηi − h(θx)(ψ_Jy)ξ|ηi, b + +

h(θx)(ψ_Jy)ξ|ηi − hψ_J(xp^∗_Jp_Jy)ξ|ηi, b + +

hψ_J(xp^∗_JpJy)ξ|ηi − hθ(xy)ξ|ηi, b < ε

3 +ε 3 +ε

3 =ε.

By Lemma 1.10, (θx)(θy) =θ(xy).

f) follows from e) ([C1], Corollary 4.4.4.8 b)).

2. EXTERIOR W^∗-TENSOR PRODUCTS

Proposition 2.1. Let E, F be C^∗-algebras, (Hi)i∈I a family of Hilbert right E-modules, (Kj)j∈J a family of Hilbert right F-modules, and

H:= |

i∈I

H_i, K:= |

j∈J

K_j, L:= |

(i,j)∈I×J

(H_i⊗K_j).

Then H⊗K ≈L. If for every (i, j)∈I×J there is a (pi, qj)∈PrE×PrF with H_i=p_iE andK_j =q_jF then

L≈ |

(i,j)∈I×J

((p_i⊗q_j)(E⊗F)).

Proof. Forξ⁰, ξ⁰⁰∈H, η⁰, η⁰⁰∈K, ξ⁰⊗η⁰

ξ⁰⁰⊗η⁰⁰

= ξ⁰

ξ⁰⁰

⊗ η⁰

η⁰⁰

=

X

i∈I

ξ⁰_i ξ_i⁰⁰

⊗

X

j∈J

η⁰_j η_j⁰⁰

=

= X

(i,j)∈I×J

( ξ_i⁰

ξ⁰⁰_i

⊗ η_j⁰

η⁰⁰_j

) = X

(i,j)∈I×J

ξ_i⁰⊗η_j⁰

ξ⁰⁰_i ⊗η_j⁰⁰ . Thus the linear map

HK →L, ξ⊗η 7→(ξi⊗ηj)_(i,j)∈I×J

is well-defined and preserves the scalar products, so it can be extended to a linear map H⊗K →L preserving the scalar products. Since this map is obviously surjective, it is an isomorphism.

The last assertion is obvious.

Corollary 2.2. Let E, F be W^∗-algebras, (H_i)i∈I a family of selfdual Hilbert right E-modules,(K_j)j∈J a family of selfdual Hilbert right F-modules,

H := |

i∈I

Hi, K := |

j∈J

Kj,

He :=

W

|

i∈I

H_i, Ke :=

W

|

j∈J

K_j, L:=

W

|

(i,j)∈I×J

(H_i⊗K¯ _j).

Then L is isomorphic to the extension of H⊗K to a selfdual Hilbert right E⊗F¯ -module, HK is dense in LL¨, and He⊗¯Ke ≈L. If for every i∈I and j ∈J there is a(p_i, q_j)∈PrE×PrF withH_i=p_iE and K_j =q_jF, then

L≈

W

|

(i,j)∈I×J

((pi⊗qj)(E⊗F¯ )).

Proof. By Proposition 2.1,

H⊗K ≈ |

(i,j)∈I×J

(H_i⊗K_j)

so the first assertion follows from Proposition 1.8. By Corollary 1.5 a), it follows that (HK)^# is dense in L^#_¨

L and that L is isomorphic to the extension of He ⊗Ke to a Hilbert rightE⊗F¯ -module, i.e.,He⊗¯Ke ≈L. The last assertion is obvious.

The next proposition will not be used in the sequel.

Proposition 2.3. Let E, F beW^∗-algebras, (H_i)i∈I a family of selfdual Hilbert right E-modules,(Kj)j∈J a family of selfdual Hilbert right F-modules,

H := |

i∈I

H_i, K := |

j∈J

K_j, L:=

W

|

(i,j)∈I×J

(H_i⊗K¯ _j).

Then H¨ K¨ is dense in L¨_L.

Proof. Let us denote by ¨HK¨ the closure of ¨HK¨ in ¨L_L. Take (a, b)∈ E¨×F¨, (i0, j0)∈I×J,ζ0 ∈Hi0⊗K¯ _j0, and define ¯ζ0 ∈L by

( ¯ζ₀)_i,j :=

ζ0 if (i, j) = (i0, j0) 0 if (i, j)6= (i₀, j₀) .

Further, let (λt)t∈T be a finite family in L and ε > 0. By [C2], Proposition 1.3 f₃), there is ζ ∈H_i0 K_j0 with

ζ−ζ0,z ^}| { (a^∗⊗b^∗,(λt)i0,j0)

< ε for every t∈T. Define ¯ζ ∈L by

ζ¯ij :=

ζ if (i, j) = (i0, j0) 0 if (i,j)6= (i₀, j0) . Then, for t∈T,

λt,z ^}| {

(a⊗b,ζ)¯ −z ^}| { (a⊗b,ζ¯0)

= λt

ζ¯−ζ¯0

, a⊗b =

=

ζ¯−ζ¯0|λt

, a^∗⊗b^∗ =

hζ−ζ0|(λt)i0,j0i, a^∗⊗b^∗ =

ζ−ζ₀,z ^}| { (a^∗⊗b^∗,(λ_t)_i0,j0)

< ε.

Since

^ z }| {

(a⊗b,ζ)¯ ∈H¨ K, we have¨

^ z }| {

(a⊗b,ζ¯0)∈H¨ K.¨ Letc∈

..

z }| {

E⊗F¯ . There is a finite family (a_s, b_s)s∈S in ¨E×F¨ such that

X

s∈S

a_s⊗b_s−c

< ε.

Put

d:=X

s∈S

as⊗bs.

By the above, (d, ζ^₀)∈H¨ K. By Lemma 1.2 a), c),¨ kz ^}| {

(d−c,ζ¯₀)k ≤ kd−ck¹²p_|d−c|( ¯ζ₀)≤ kd−ck kζ¯₀k ≤εkζ¯₀k.

It follows (c,^ζ¯0)∈H¨ ×K.¨

Let nowζ1 ∈L. By [C1], Theorem 5.6.3.13 f) (and [C1], Theorem 5.6.4.6 f)), there is a finite subsetA of I×J such that

ζ₁−ζ_1A,(c^^∗, λ_t) < ε for every t∈T, where

ζ_1A:=

(ζ₁)_ij if (i, j)∈A

0 if (i, j)∈(I×J)\A . By the above, (c, ζ^1A)∈H¨ ×K. For¨ t∈T,

λ_t,(c, ζ^₁)−(c, ζ^_1A) =

λ_t,z ^}| { (c, ζ₁−ζ_1A)

=

hλ_t|ζ₁−ζ_1Ai, c =

=

hζ₁−ζ1A|λti, c^∗ =

ζ1−ζ1A,(c^^∗, λt) < ε.

Thus,(c, ζ^1)∈H¨ K¨ and ¨HK¨ = ¨L.

Theorem 2.4. ([C2], Theorem 2.4) If E, F are W^∗-algebras and if H (resp. K) is a selfdual Hilbert right E-module(resp. F-module) then

L_E(H) ¯⊗L_F(K)≈ L_E⊗F¯ (H⊗K).¯

Proof. By [C1], Proposition 5.6.4.10 a), there are families (p_i)i∈I and (qj)j∈J in PrE and PrF, respectively, such that

H≈

W

|

i∈I

(piE), K≈

W

|

j∈J

(qjF).

By Corollary 2.2,

H⊗K¯ ≈ ^W|

(i,j)∈I×J

((p_i⊗q_j)(E⊗F¯ ))

and by Corollary 1.5 a1) ⇒ a2), (H⊗K)^# is dense in (H⊗H)¯ ^#..

z }|{ H⊗K¯

.

We may assume thatE andF are von Neumann algebras on the Hilbert spaces L and M respectively. Then E⊗F¯ is a von Neumann algebra on the Hilbert space L⊗M. We put

Le:= |

i∈I

pi(L), Mf:= |

j∈J

qj(M).

By Theorem 1.12 f),L_E(H),L_F(K), andL_E⊗F¯ (H⊗K) may be identified with¯ von Neumann algebras on L,e Mf, and Le⊗M, respectively. We havef

L_E(H)⊗ L_F(K)⊂ L(L)e ⊗ L(Mf)⊂ L(Le⊗M).f By Proposition 1.4 e) (and Corollary 1.5 a1) ⇒ a2)),

LE⊗F(H⊗K)⊂ L_E⊗F¯ (H⊗K)¯ and so (by [L], page 36)

L_E(H)⊗ L_F(K)⊂ LE⊗F(H⊗K)⊂ L_E⊗F¯ (H⊗K)¯ ⊂ L(Le⊗Mf).

By Corollary 1.5 b), K_E⊗F(H⊗K) is dense inL_E⊗F¯ (H⊗K)¯ ^...

z}|{ H⊗K¯

. Since KE⊗F(H⊗K) =K_E(H)⊗ K_F(K)⊂ L_E(H)⊗ L_F(K) ([L], page 37), L_E(H)⊗ L_F(K) is dense in L_E⊗F¯ (H⊗K)¯ ^...

z}| { H⊗H¯

, so by [C1], Corollary 4.4.4.12 a),

L_E(H) ¯⊗L_F(K)≈ L_E⊗F¯ (H⊗K).¯

Proposition 2.5. Let E, F be W^∗-algebras, H (resp. K) a selfdual Hilbert right E-module (resp. F-module), and E₀ (resp. F₀) a W^∗-subalgebra of L_E(H) (resp.L_F(K)). Consider the inclusions (Theorem 2.4)

E0⊗F0 ⊂ L_E(H)⊗ L_F(K)⊂ L_E(H) ¯⊗L_F(K)≈ L_E⊗F¯ (H⊗K).¯ Then E₀⊗F¯ ₀ is isomorphic to the closure ofE₀⊗F₀ in

(L_E⊗F¯ (H⊗K))¯ ^...

z }| { H⊗K¯ .

Proof. By [C1], Proposition 5.6.4.10 a), there is a family (pi)i∈I in PrE and a family (q_j)j∈J in PrF such that

H≈ ^W|

i∈I

(p_iE), K≈ ^W|

j∈J

(q_jF).

Assume E (resp. F) is a von Neumann algebra on the Hilbert space M (resp. N) and put

P := |

i∈I

pi(M), Q:= |

j∈J

qj(N).