HILBERT RIGHT SEMIFINITE W

HILBERT RIGHT SEMIFINITE W

-MODULES

CORNELIU CONSTANTINESCU

Let E be a semifinite W^∗-algebras. We define the Hilbert dimension for every selfdual Hilbert rightE-module in such a way that two suchE-modules are iso- morphic iff their Hilbert dimensions coincide.

AMS 2000 Subject Classification: 46L08.

Key words: selfdual Hilbert rightW^∗-module.

INTRODUCTION

Every Hilbert space H has a Hilbert dimension, say DimH, which is the cardinal number of the orthonormal bases of H. Two Hilbert spaces are isomorphic iff their Hilbert dimensions coincide and, for every cardinal number ℵ,Diml²(ℵ) =ℵ.Moreover, if H and K are Hilbert spaces, then

Dim (H| K) = DimH+ DimK, Dim (H⊗K) = (DimH) (DimK), and the statements below are equivalent.

a) DimH≤DimK.

b) There is a Hilbert space L such thatK =H| L.

c) There is a surjective operator K →H.

d) There is an injective operator H→K.

Let E be a semifinite W^∗-algebra. We define the Hilbert dimension for every selfdual Hilbert right E-module and try to extend the above landscape from classical Hilbert spaces to selfdual Hilbert right E-modules.

In general, we use the notation and terminology of [C1]. For W^∗-tensor products of W^∗-algebras we use [T], for tensor products of Hilbert right C^∗- modules we use [La], and for W^∗-tensor products of selfdual Hilbert right W^∗-modules we use [C2].

We now give a list of some notation used in this paper.

1. Kdenotes the field of real numbers or the field of complex numbers.

The whole theory is developed in parallel for the real and complex case, but the proofs coincide. N denotes the set of natural numbers (0 6∈ N) and for

MATH. REPORTS11(61),2 (2009), 75–137

everyn∈Nwe putN_n:={k∈N|k≤n}.R₊denotes the set of positive real numbers (0∈R₊).

2. IfX is a set, then for every subsetA ofX we denote 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. If X, Y are sets,

f :X→Y, x7→f(x)

a map, and Aa subset of X, thenf|A denotes the restriction off toA, i.e., f|A:A→Y, x7→f(x).

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

5. For ordinal numbers we adopt von Neumann’s point of view: if α, β are ordinal numbers then α∈β ⇔α < β. In other words, an ordinal number β is just the set of all ordinal numbers strictly less than β.

6. In ordered sets, ∧and ∨ denote the infimum and supremum, respec- tively.

7. If E is a Banach space, thenE^#denotes the unit ball of E:

E^#:={x∈E | kxk ≤1}.

IfE has a unique (or a distinguished) predual, then we denote by ¨E this pre- dual and byE_E¨ the vector spaceE endowed with the locally convex topology of pointwise convergence on ¨E. If (x_i)_i∈I is a summable family in E_E¨, then we denote by

E¨

P

i∈I

x_i its sum. It is characterized by the relation

a∈E¨ ⇒

*XE¨ i∈I

xi, a +

=X

i∈I

hxi, ai.

8. If H is a Hilbert space and ξ, η ∈H, then we denote by hξ|ηi their scalar product and puth · |ηiξ:H →H,ζ 7→ hζ|ηiξ;L(H) denotes theW^∗- algebra of operators H →H and L^′(H) its predual (which may be identified with the set of nuclear operators of L(H)).

9. If I is a set, thenl²(I) denotes the Hilbert space of square summable families inKindexed byI and (e_i)_i∈I denotes its canonical orthonormal basis.

For i, j∈I,δ_ij denotes the Kronecker’s symbol δij :=

( 1 ifi=j 0 ifi6=j .

10. For every C^∗-algebra E we denote by ReE the set of selfadjoint elements of E. A Gelfand C^∗-algebra is a commutative C^∗-algebra E such

that in the real case all its elements are selfadjoint. We denote by σ(E) its spectrum and byC(σ(E)) theC^∗-algebra of continuous scalar valued functions on σ(E). The Gelfand transform

E → C(σ(E)), x7→xb is an isomorphism of C^∗-algebras.

11. Let E be a W^∗-algebra. We denote by 1_E its unit and by PrE the complete lattice of orthogonal projections ofE. For p, q∈PrE we put p∼q (resp. pq) if there isx∈E with

x^∗x=p, xx^∗=q (resp.xx^∗ ≤q).

The notation∼ and will be used also forC^∗-algebras. We put E^c :={x∈E |y∈E ⇒xy=yx}.

AC^∗-subalgebraF ofEis calledW^∗-subalgebra if the inclusion mapFF¨ →EE¨

is continuous; ¨E+ denotes the convex cone of positive elements of the predual of E.

12. LetE be aC^∗-algebra andH, K Hilbert rightE-modules. We denote by L(H, K) the Banach space of operators H→K, by LE(H, K) its Banach subspace of adjointable operators, by LbE(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 :=LbE(H, E) and

h · |ξ i:H→E, η7→ hη|ξi

for every ξ ∈H. If for every u∈Hb there is ξ ∈H withu:= h · |ξ i,then we say that H is selfdual.

13. Let Ebe aC^∗-algebra. ForH, K Hilbert rightE-modules we use the notation H≈K for “H andK are isomorphic”.

14. LetE be aW^∗-algebra andHa selfdual Hilbert rightE-module. For a∈E¨ and ξ, η ∈H we put

(a, ξ) :] H→K, ζ 7→ h hζ|ξi, ai, (a, ξ, η) :^ LE(H)→K, u7→ h huξ|ηi, ai. Then H and LE(H) have preduals which are generated by

{(a, ξ)] |(a, ξ)∈E¨×H} and {(a, ξ, η)^ |(a, ξ, η)∈E¨×H×H}, respectively. In this case we denote by H^... the predual of LE(H). If p ∈ PrLE(H) then pH := Imp = {pξ | ξ ∈ H} is a closed Hilbert right E- submodule of H. If K is a closed Hilbert right E-submodule of H, then

we put

K^⊥:={ξ ∈H |η∈H ⇒ hξ|ηi= 0}.

A Fourier set of H is a subsetA of H such that

ξ∈A⇒ hξ|ξi ∈PrE, ξ, η ∈A, ξ 6=η⇒ hξ|ηi= 0.

A Fourier basis ofH is a maximal Fourier set of H.

15. If E is a W^∗-algebra and (H_i)_i∈I is a family of selfdual Hilbert right E-modules, then we put

|

i∈I

H_i :=

ξ ∈Y

i∈I

H_i |the family (hξ_i|ξ_i i)_i∈I is summable inE

, W|

i∈I

H_i:=

ξ ∈Y

i∈I

H_i |the family (hξ_i|ξ_i i)_i∈I is summable inE_E¨

. 16. ⊙denotes the algebraic tensor product of vector spaces.

17. If E, F areW^∗-algebras andH (resp. K) is a selfdual Hilbert right E- (resp. F-) module, then we denote byH⊗K theW^∗-tensor product ofH and K, which is a selfdual Hilbert right E⊗F-module ([C2], Definition 2.3).

18. HK for Hilbert rightC^∗-modules H, K: Definition 2.3.

19. H(E), class of selfdual Hilbert rightE-modules: Definition 2.10.

20. Γ(E), 0^Γ(E)∈Γ(E),H_p forp∈Γ(E): Definition 3.4.

21. Semi-abelian orthogonal projection, semi-abelianW^∗-algebra, type I W^∗-algebra: beginning of Section 4.

22. Cd,Cd0, classes of (mainly) cardinal numbers with order relation ≤ and operations +,−: Definition 4.1.

23. Hilbert dimension of Hilbert right W^∗-modules: Theorems 4.9, 5.17.

24. Type II W^∗-algebra: beginning of Section 5.

25. ∆(E), 0^∆(E)∈∆(E), HZ forZ ∈∆(E): Definition 5.3.

1. MATRIX REPRESENTATIONS OF OPERATORS

We start with

Lemma 1.1. Let E be a W^∗-algebra, (H_i)_i∈I a family of selfdual Hilbert right E-modules, andH:= |

i∈I

H_i, K:= ^W|

i∈I

H_i ([C1],Proposition5.6.4.1 c), [C1], Proposition 5.6.4.6 b)). For every η ∈ K we have h · |ηi |H ∈ Hb and the map

K →H,b η7→ h · |η i |H is an isomorphism of Hilbert right E-modules.

Proof. It is obvious that h · |ηi |H ∈Hb for every η ∈K. Moreover, for η∈K, x∈E,and ξ ∈H we have

(h · |ηxi)ξ =hξ|ηxi=x^∗hξ|ηi= (h · |ηix)ξ, so that

h · |ηxi=h · |ηix.

Let P_f(I) be the set of finite subsets of I,F its upper section filter and for everyJ ∈P_f(I) lete_J be its characteristic function. For η∈K we have

limJ,Fhη−ηe_J|η−ηe_J i= lim

J,F E¨

X

i∈I\J

hη_i|η_i i= 0 inE_E¨. By ([C1], Theorem 5.6.2.11 f)),

h h · |ξi | h · |ζ i i=hξ|ζ i for (ξ, ζ)∈H×K. By the above,

limJ,Fh h · |ηi − h · |ηe_J i | h · |ζ i i= 0 forη, ζ ∈K inE_E¨, so that

h h · |ηi | h · |ζ i i= lim

J,Fh h · |ηe_J i | h · |ζ i i= lim

J,Fhηe_J|ζi=hη|ζ i. Thus, we have to prove the surjectivity of the map only.

Let ϕ ∈ H. For everyb i ∈ I, we identify H_i with a Hilbert right E- submodule of H in a natural way. Since H_i is selfdual, there isη_i ∈H_i such that ϕ(ξ_i) =hξ_i|η_ii for everyξ_i ∈H_i. LetJ be a finite subset of I and

ηJ :=

( ηi ifi∈J 0 ifi∈I\J . Then

ϕ(η_J) =X

i∈J

ϕ((η_J)_i) =X

i∈J

hη_i|η_ii=hη_J|η_J i, kηJk² =k hη_J|η_J i k=kϕ(ηJ)k ≤ kϕk kηJk, kηJk ≤ kϕk.

Thus, η := (η_i)_i∈I ∈K and ϕ(ξ_i) =hξ_i|η_i i = (h · |ηi)ξ for every i∈I and ξ_i ∈H_i. It follows that ϕ=h · |ηi |H, hence the map is surjective.

Lemma 1.2. Let E, F be C^∗-algebras, (H_i)_i∈I a family of Hilbert right E-modules, (K_j)_j∈J a family of Hilbert right F-modules, and

H := |

i∈I

H_i, K:= |

j∈J

K_j.

Then the linear map H⊙K → |

(i,j)∈I×J

(H_i⊗K_j), ξ⊗η7→(ξ_i⊗η_j)_(i,j)∈I×J can be extended to an isomorphism

H⊗K → |

(i,j)∈I×J

(H_i⊗K_j).

of Hilbert right E⊗F-modules.

Proof. Let us denote by ϕ the first map and let (ξ^′_s, η^′_s)_s∈S, (ξ^′′_t, η_t^′′)_t∈T be finite families in H×K. Put

ζ^′:=X

s∈S

ξ^′_s⊗η_s^′, ζ^′′:=X

t∈T

ξ^′′_t ⊗η_t^′′.

Then

ϕζ^′ ϕζ^′′

= X

(s,t)∈S×T

ϕ(ξ_s^′ ⊗η^′_s)

ϕ(ξ_t^′′⊗η^′′_t)

=

= X

(s,t)∈S×T

(ξ_si^′ ⊗η^′_sj)_(i,j)∈I×J(ξ_ti^′′ ⊗η_tj^′′)_(i,j)∈I×J

=

= X

(s,t)∈S×T

X

(i,j)∈I×J

ξ_si^′ ⊗η_sj^′

ξ^′′_ti⊗η_tj^′′

=

= X

(s,t)∈S×T

X

(i,j)∈I×J

ξ_si^′ ξ_ti^′′

⊗ η^′_sj

η_tj^′′

=

= X

(s,t)∈S×T

X

i∈I

ξ^′_siξ^′′_ti

⊗ X

j∈J

η_sj^′ η^′′_tj

=

= X

(s,t)∈S×T

ξ_s^′ ξ_t^′′

⊗ η_s^′

η_t^′′

= X

(s,t)∈S×T

ξ_s^′ ⊗η_s^′

ξ_t^′′⊗η_t^′′

=

=

*X

s∈S

ξ_s^′ ⊗η_s^′

X

t∈T

ξ_t^′′⊗η^′′_t +

= ζ^′

ζ^′′

. Thus,ϕmay be extended by continuity to a map

H⊗K → |

(i,j)∈I×J

(H_i⊗K_j)

which is obviously surjective and so it is an isomorphism.

Lemma 1.3. Let E and F be W^∗-algebras and G a W^∗-subalgebra ofF. Then E⊗G is the closure of E⊗G in (E⊗F)z }| {_¨

E⊗F .

Proof. Let H and K be Hilbert spaces such that E and F are von Neumann algebras on H and K, respectively. Then E⊗G and E⊗F are the closures of E ⊗G and E ⊗F in L(H⊗K)_L^′_(H⊗K), respectively. Since L(H⊗K)_L^′_(H⊗K) induces on E⊗F the topology (E⊗F)z }| {_¨

E⊗F

, E⊗G is the closure of E⊗Gin (E⊗F)z }| {_¨

E⊗F

.

The next result is nothing else but ([T], Section IV.1) generalized to selfdual Hilbert right W^∗-modules.

Theorem 1.4. Let E be a W^∗-algebra, H a selfdual Hilbert right E- module, I a set, K :=l²(I), and L:=H⊗K ([C2], Definition 2.3), which is a selfdual Hilbert right E⊗K-module (E⊗K=E).

a) For every ξ ∈ ^W|

i∈I

H, the family (ξ_i⊗e_i)_i∈I is summable in L_L¨ and the map

W

|

i∈I

H → L, ξ 7→

L¨

P

i∈I

ξi⊗ei is a (canonical) isomorphism of Hilbert right E-modules.

b) IfM is a Hilbert rightE-module andu∈ LE(L, M)withu(ξ⊗e_i) = 0 for all (ξ, i)∈H×I, then u= 0.

c) For everyi∈I putu_i :H →L,ξ 7→ξ⊗e_i.Then, for i, j∈I, ξ ∈H, and η∈K we have

u_i ∈ L_E(H, L), u^∗_i(ξ⊗e_j) =δ_ijξ, u^∗_i(ξ⊗η) =hη|e_i iξ, u^∗_iu_j =δ_ij1_H, u_iu^∗_j = 1_H ⊗(h · |e_j ie_i), X

i∈I

u_iu^∗_i = 1_L (pointwise).

d) For everyx∈ LE(L)andi, j∈I, putxij :=u^∗_ixuj.Then, fori, j∈I, x, y∈ LE(L), z∈ LE(H), and w∈ L(K) we have

xij ∈ LE(H), x^∗_ij = (x^∗)ji, (xy)ij =X

k∈I

x_iky_kj (pointwise), (z⊗1K)ij =δijz, (1H ⊗w)ij =hwej|eii1H,

(z⊗w)_ij =hwe_j|e_i iz, u_izu^∗_j = (z⊗1_K)u_iu^∗_j =z⊗(h · |e_j ie_i), x=X

i∈I

X

j∈I

u_ix_iju^∗_j =X

i∈I

X

j∈I

x_ij ⊗(h · |e_j ie_i) (pointwise), and the map LE(L)^#...

L → LE(H)^#...

H,x7→x_ij is continuous.

e) {u_iu^∗_j |i, j∈I}^c ={x⊗1_K |x∈ L_E(H)}.

f) If F is a W^∗-subalgebra of LE(H), then

f₁) F⊗ L(K) ={x∈ LE(L)|i, j ∈I ⇒x_ij ∈F}; f₂) (F⊗K)^c =F^c⊗ L(K);

f₃) (F⊗ L(K))^c =F^c⊗K.

Proof. a) By Lemma 1.2 the linear map H⊙l²(I)→ |

i∈I

H, η⊗ζ 7→(hζ|e_iiη)_i∈I can be extended to an isomorphism ψ : H⊗l²(I) → |

i∈I

H of Hilbert right E-modules. Then

ψb: \|

i∈I

H→

z \}| {

H⊗l²(I), u7→u◦ψ

also is an isomorphism of Hilbert rightE-modules ([C1], Theorem 5.6.2.11 g)).

If we put G:= ^W|

i∈I

H, then, by Lemma 1.1, the map ϕ:G→ \|

i∈I

H, ξ7→ h · |ξi |( |

i∈I

H)

is an isomorphism of Hilbert right E-modules. By ([C2], Definition 2.3) and ([C2], Proposition 1.3 f)), we haveL=

z \}| {

H⊗l²(I), so thatψb◦ϕ:G→L is an isomorphism of Hilbert right E-modules and thus the map G_G¨ →L_L¨ defined by ψb◦ϕalso is an isomorphism.

Let ξ ∈ G and for every i ∈ I put ˙ξ_i := (δ_ijξ_i)_j∈I ∈ G. For (η, ζ) ∈ H×l²(I) we have

(ψϕb ξ˙i)(η⊗ζ) = (ϕξ˙i)ψ(η⊗ζ) =D

ψ(η⊗ζ) ξ˙i

E

=

=h(hζ|e_j iη)_j∈I|(δ_ijξ_i)_j∈Ii=hζ|e_ii hη|ξ_i i=hη⊗ζ|ξ_i⊗e_i i, so that ([C1], Theorem 5.6.2.11 e)), ψϕb ξ˙_i = ξ_i ⊗e_i. By ([C1], Proposition 5.6.4.6 c)), ( ˙ξ_i)_i∈I is summable inG_G¨ and

G¨

P

i∈I

ξ˙_i =ξ. By the above, (ψϕb ξ˙_i)_i∈I is summable in L_L¨ and

L¨

P

i∈I

ψϕb ξ˙_i=ψϕξ.b Hence (ξ_i⊗e_i)_i∈I is summable inL_L¨ and

L¨

P

i∈I

ξ_i⊗e_i=ψϕξ.b Thus, the map

W

|

i∈I

H→L, ξ 7→

L¨

X

i∈I

ξi⊗ei

is nothing else but the isomorphism ψb◦ϕof Hilbert right E-modules.

b) By ([C1], Proposition 5.6.3.4 c)), the map L_L¨ → M_M¨, ζ 7→ uζ is continuous and the assertion follows from a).

c) For ζ ∈H we have

hu_iξ|ζ⊗e_j i=hξ⊗e_i|ζ⊗e_j i=δ_ijhξ|ζ i=hξ|δ_ijζ i. By a),u_i∈ LE(H, L) and

u^∗_i(ζ⊗ej) =δijζ, u^∗_i(ξ⊗η) =X

j∈I

u^∗_i(ξ⊗ hη|ej iej) =X

j∈I

δijhη|ej iξ =hη|ei iξ.

It follows that

u^∗_iu_jξ=u^∗_i(ξ⊗e_i) =δ_ijξ, u^∗_iu_j =δ_ij1_H,

u_iu^∗_j(ξ⊗η) =u_ihη|e_j iξ =ξ⊗ hη|e_j ie_i = (1_H⊗(h · |e_j ie_i))(ξ⊗η), so that, by b),

uiu^∗_j = 1H ⊗(h · |ej iei).

Since P

i∈I

h · |ei iei = 1K pointwise, we have X

i∈E

u_iu^∗_i = 1_L pointwise.

d) By c),

x_ij ∈ LE(H), x^∗_ij =u^∗_jx^∗u_i= (x^∗)_ji, (xy)_ij =u^∗_ixyu_j =u^∗_ixX

k∈K

u_ku^∗_kyu_j =X

k∈J

x_iky_kj (pointwise).

For ξ∈H, again by c),

(z⊗1_K)ijξ =u^∗_i(z⊗1_K)ujξ=u^∗_i(z⊗1_K)(ξ⊗ej) =u^∗_i((zξ)⊗ej) =δijzξ, (z⊗1_K)_ij =δ_ijz,

(1_H⊗w)_ijξ =u^∗_i(1_H ⊗w)u_jξ=u^∗_i(1_H ⊗w)(ξ⊗e_j) =

=u^∗_i(ξ⊗(we_j)) =hwe_j|e_i iξ, (1H ⊗w)ij =hwej|ei i1H, (z⊗w)_ij = ((z⊗1_K)(1_H ⊗w))_ij =X

k∈I

(z⊗1_K)_ik(1_H ⊗w)_kj =

=X

k∈I

δ_ikzhwe_j|e_ki=hwe_j|e_i iz.

For k∈I, by c) we have

u_izu^∗_j(ξ⊗e_k) =u_izδ_jkξ =δ_jk(zξ)⊗e_i =

= (zξ)⊗ he_k|e_j ie_i = (z⊗(h · |e_j ie_i))(ξ⊗e_k)

so, by b), u_izu^∗_j =z⊗(h · |e_j ie_i.By c) we have x= X

i∈I

u_iu^∗_i

x X

j∈I

u_ju^∗_j

=X

i∈I

X

j∈I

u_iu^∗_ixu_ju^∗_j =

=X

i∈I

X

j∈I

u_ix_iju^∗_j =X

i∈I

X

j∈I

x_ij ⊗(h · |e_j ie_i) (pointwise).

For (a, ξ, η)∈E¨×H×H we have Dx_ij,(a, ξ, η)^ E

=h hx_ijξ|η i, ai=h hu^∗_ixu_jξ|ηi, ai=

=h hxujξ|uiηi, ai=D x,

z ^}| { (a, ujξ, uiη)E

, so that the map

LE(L)^#...

L → LE(H)^#...

H, x7→xij

is continuous.

e) Let x∈ LE(H) andi, j∈I. By c) and d), u_iu^∗_j(x⊗1_K) =u_iu^∗_j(x⊗1_K)X

k∈I

u_ku^∗_k =X

k∈I

u_i(x⊗1_K)_jku^∗_k=

=X

k∈I

δ_jku_ixu^∗_k=u_ixu^∗_j = (x⊗1_K)u_iu^∗_j,

i.e., x⊗1_K ∈ {uiu^∗_j |i, j ∈I}^c and {x⊗1_K |x∈ L_E(H)} ⊂ {uiu^∗_j |i, j∈I}^c. Let y∈ {uiu^∗_j |i, j∈I}^c. Fori, j, k, l∈I, by c),

δ_kly_ij =y_iju^∗_ku_l=u^∗_iyu_ju^∗_ku_l=u^∗_iu_ju^∗_kyu_l=δ_ijy_kl.

It follows that i 6= j ⇒ y_ij = 0, y_ii = y_kk. Putting x := y_ii for an i∈ I we have y_ij =δ_ijx and, by d),

y=X

i∈I

X

j∈I

y_ij ⊗(h · |e_j ie_i) =X

i∈I

X

j∈I

(δ_ijx)⊗(h · |e_j ie_i) =

=X

i∈I

x⊗(h · |e_i ie_i) =x⊗1_K,

y∈ {x⊗1K |x∈ LE(H)}, {uiu^∗_j |i, j∈I}^c ={x⊗1K |x∈ LE(H)}.

f1) Let (x, y)∈F× L(K) and i, j∈I. By d), σij :LE(L)^#...

L → LE(H)^#...

H, x7→xij

is continuous andσ_ij(x⊗y) =hye_j|e_iix∈F.Thus,σ_ij(F⊗L(K))⊂F.Since F⊗ L(K) is the closure ofF⊗ L(K) inLE(H)⊗ L(K) =L(L)^..._L (Lemma 1.3, [C2] Theorem 2.4 d)), we have

(F⊗ L(K))z }|¨ { F⊗ L(K)

= (F⊗ L(K))^...

L

([C1], Corollary 4.4.4.9)). Let z∈(F⊗ L(K))^#. Since (F ⊗ L(K))^# is dense in (F⊗ L(K))^#...

L ([C1], Corollary 6.3.8.7)), there is a filterF on (F ⊗ L(K))^# converging to zin (F⊗ L(K))^#...

L . By the above, z_ij =σ_ijz= lim

w,Fσ_ijw∈F sinceF is closed inL(H)_H^.... Thus,

F⊗ L(K)⊂ {x∈ LE(L)|i, j∈I ⇒x_ij ∈F}.

Now, let x∈ LE(L) withx_ij ∈F for all i, j∈I. By d), x=X

i∈I

X

j∈I

xij ⊗(h · |ej iei) pointwise, hence with respect to the topology of LE(L)^...

L. Thus, x∈F⊗ L(K), {x∈ L_E(L)|i, j∈I ⇒xij ∈F} ⊂F⊗ L(K).

f₂) Letx∈(F ⊗K)^c. Fori, j∈I and y∈F, by c) and d), we have xijy=X

k∈I

xikδkjy=X

k∈I

xik(y⊗1K)kj = (x(y⊗1K))ij =

= ((y⊗1_K)x)_ij =X

k∈I

(y⊗1_K)_ikx_kj =X

k∈I

δ_ikyx_kj =yx_ij, i.e., xij ∈F^c. By f1),x∈F^c⊗ L(K), (F ⊗K)^c ⊂F^c⊗ L(K).

Now, let y∈F^c and z∈ L(K). Forx∈F we have

(y⊗z)(x⊗1_K) = (yx)⊗z= (xy)⊗z= (x⊗1_K)(y⊗z), so that

y⊗z∈(F⊗K)^c, F^c⊗ L(K)⊂(F⊗K)^c ⊂ LE(H)⊗ L(K).

By Lemma 1.3, we then haveF^c⊗ L(K)⊂(F⊗K)^c. f₃) Letx∈F^c and (y, z)∈F ⊗ L(K). Then

(x⊗1_K)(y⊗z) = (y⊗z)(x⊗1_K), so that

x⊗1_K ∈(F⊗ L(K))^c, x⊗1_K ∈(F⊗ L(K))^c, F^c⊗K⊂(F⊗ L(K))^c.

Letx∈(F⊗ L(K))^c. By c) and e), there isx₀ ∈ L_E(H) withx=x₀⊗1_K. For y∈F we have

(x₀y)⊗1_K = (x₀⊗1_K)(y⊗1_K) = (y⊗1_K)(x₀⊗1_K) = (yx₀)⊗1_K, so that

x0y=yx0, x0∈F^c, x∈F^c⊗K, (F⊗ L(K))^c ⊂F^c⊗K. Proposition 1.5. Use the notation of Theorem1.4and assumeH =E.

Let (p_i)_i∈I, (q_i)_i∈I, and (r_i)_i∈I be families in PrE, and P := ^W|

i∈I

p_iE, Q:= ^W|

i∈I

q_iE, R:= ^W|

i∈I

r_iE, S:= ^W|

i∈I

E, ϕ:P →S, ψ:Q→S, θ:R→S

the inclusion maps. Identify S with L using the map of Theorem1.4 a).

a) ϕ∈ LE(P, S) and ϕ^∗ξ= (p_iξ_i)_i∈I for everyξ∈S. In particular,ϕ^∗ϕ is the identity map of P.

b) Let v ∈ LE(P, Q) and x := ψvϕ^∗. Then x ∈ LE(L), x^∗ = ϕv^∗ψ^∗, kxk=kvk, andx_ij ∈q_iEp_j for alli, j∈I.

c) The map

L_E(P, Q)→ {x∈ L_E(L)|i, j ∈I ⇒x_ij ∈q_iEp_j}, v7→ψvϕ^∗ is linear and bijective.

d) If v∈ LE(P, Q), w∈ LE(Q, R), then

(θwψ^∗)◦(ψvϕ^∗) =θ(wv)ϕ^∗.

e)Letv∈ LE(P, Q) andx:=ψvϕ^∗.Thenvis an isomorphism of Hilbert right E-modules iff

j, l∈I ⇒

E¨

X

m∈I

x^∗_mjx_ml =δ_jlpj, i, k ∈I ⇒

E¨

X

n∈I

xinx^∗_kn=δ_ikqi.

f) Let r∈PrE such thatp_i =rp_ir, q_i =rq_ir for alli∈I, and putF :=

rEr. If P and Q are isomorphic, then the Hilbert right F-modules ^W|

i∈I

p_iF, and

W

|

i∈I

q_iF are isomorphic, too.

g) Assume P and Qare isomorphic and E is aW^∗-subalgebra of aW^∗- algebraF. Then the Hilbert right F-modules

W

|

i∈I

piF and

W

|

i∈I

qiF are isomor- phic, too.

Proof. a) Easy.

b) By a), x ∈ LE(L). The relations x^∗ = ϕv^∗ψ^∗, and kxk = kvk are easily checked. For ξ ∈E, by a),

ϕ^∗u_jp_jξ=ϕ^∗((p_jξ)⊗e_j) = (p_jξ)⊗e_j =ϕ^∗(ξ⊗e_j) =ϕ^∗u_jξ, so that ϕ^∗u_jp_j =ϕ^∗u_j and x_ijp_j =x_ij.By Theorem 1.4 d),

x^∗_ij = (x^∗)_ji= (x^∗)_jiq_i =x^∗_ijq_i, x_ij =q_ix_ij, x_ij ∈q_iEp_j. c) If ψvϕ^∗ = 0 then, by a),

v= (ψ^∗ψ)v(ϕ^∗ϕ) =ψ^∗(ψvϕ^∗)ϕ= 0, so that the map is injective.

Now, let x ∈ LE(L) with x_ij ∈q_iEp_j for all i, j ∈I and letξ ∈L. For i, j∈I we have

(xij ⊗(h · |ej iei))ξ = (xij⊗(h · |ej iei))

L¨

X

k∈I

ξ_k⊗e_k=

=

L¨

X

k∈I

x_ijξ_k⊗ he_k|e_j ie_i=x_ijξ_j⊗e_i. By Theorem 1.4 d),

xξ= X

i,j∈I

x_ij⊗(h · |e_j ie_i)

ξ= X

i,j∈I

(x_ij⊗(h · |e_j ie_i))ξ= X

i,j∈I

x_ijξ_j⊗e_i, so that xξ=ψ^∗xξ∈Q,xϕ^∗ξ=xξ. Put

v:P →Q, ξ7→xξ.

Then v∈ LE(P, Q) and

ψvϕ^∗ξ=ψxϕ^∗ξ=ψxξ=ψψ^∗xξ=xξ, ψvϕ^∗ =x.

Thus, the map is surjective.

d) follows from a).

e) Ifvis an isomorphism, thenv^∗v= 1_P,vv^∗ = 1_Qand the conditions are fulfilled by d) and Theorem 1.4 d). Conversely, if the conditions are fulfilled then, by d),v^∗v= 1_P,vv^∗ = 1_Q, i.e., v is an isomorphism.

f) Let x ∈ LE(P, Q) be an isomorphism. By b), xij ∈F for all i, j ∈ I and the assertion follows from e).

g) Let v : P → Q be an isomorphism of Hilbert right E-modules and x:=ψvϕ^∗.By b) and e),x_ij ∈q_iEp_j ⊂q_iF p_j for all i, j∈I and

j, l∈I ⇒

E¨

X

m∈I

x^∗_mjx_ml =δ_jlp_j, i, k ∈I ⇒

E¨

X

n∈I

x_inx^∗_kn=δ_ikq_i.

By ([C1], Corollary 4.4.4.9)), we may replace ¨E by ¨F in the above sums and the assertion follows from c) and e) applied to F instead ofE.

2. SELFDUAL HILBERT RIGHTW^∗-MODULES

We start with

Lemma 2.1. Let E be a W^∗-algebra, (p_i)_i∈I a family in PrE, H :=

E⊗l²(I), F :=LE(H), and ϕ: ^W|

i∈I

E →H, ξ7→

H¨

X

i∈I

ξ_i⊗e_i

the canonical isomorphism of Hilbert right E-modules (Theorem 1.4 a)). For every i∈I put ω_i :=h · |e_iie_i ∈Pr(L(l²(I))).Then

p:=

F¨

X

i∈I

pi⊗ωi∈PrF and Imp=ϕ _W

|

i∈I

piE

.

Proof. (pi ⊗ωi)i∈I is a family in PrF such that (pi⊗ωi)(pj ⊗ωj) = 0 for all distincti, j ∈I, so thatpis well-defined and belongs to PrF. Fori∈I and ξ ∈ ^W|

i∈I

E, by ([C1], Proposition 5.6.3.4 c)),

(p_i⊗ω_i)ϕξ=

H¨

X

j∈I

(p_i⊗ω_i)(ξ_j⊗e_j) =

H¨

X

j∈I

(p_iξ_j)⊗(ω_ie_j) = (p_iξ_i)⊗e_i. For a∈E¨ and η∈H we have

Dpϕξ, (a, η)] E

=D

p,(a, ϕξ, η)^ E

=X

i∈I

Dp_i⊗ω_i,(a, ϕξ, η)^ E

=

=X

i∈I

D

(pi⊗ωi)ϕξ, (a, η)] E

=X

i∈I

D

(piξi)⊗ei,(a, η)] E , so that

pϕξ =

H¨

X

i∈I

(p_iξ_i)⊗e_i and Imp=ϕ _W

|

i∈I

p_iE

.

Lemma 2.2. Let E be a C^∗-algebra, H a Hilbert right E-module, p ∈ PrLE(H), and K:= Imp.

a) Kerp=K^⊥. b) K^⊥⊥=K.

c) If H is selfdual, then K is selfdual, too.

Proof. a) Take ξ ∈ Kerp. Then hξ|ηi = hξ|pηi = hpξ|ηi = 0 for η∈K, so that ξ∈K^⊥.

Conversely, assume ξ ∈K^⊥. Then 0 = hξ|ηi =hξ|pηi =hpξ|ηi for η∈K, so that pξ∈K^⊥. Since pξ ∈K, we have pξ = 0,ξ ∈Kerp.

b) K ⊂K^⊥⊥ is trivial. Let ξ ∈K^⊥⊥ and put η:=ξ−pξ ∈K^⊥⊥. Then pη=pξ−p²ξ = 0, η∈Kerp=K^⊥

by a). It follows thatη= 0 and ξ =pξ ∈K.

c) Let u ∈ LbE(K, E). Put v := u◦p ∈ LbE(H, E). Since H is selfdual there is η∈H withv=h · |ηi.For ξ∈K we have

uξ=up²ξ=vpξ=hpξ|η i=hξ|pηi, so that u=h · |pηi and K is selfdual.

Definition 2.3. LetEbe aC^∗-algebra andH, KHilbert rightE-modules.

Write HK if there is a Hilbert rightE-moduleL such thatK ≈H| L.

Proposition 2.4. Let E be a C^∗-algebra, H a Hilbert right E-module, F :=LE(H), p, q∈PrF, and i:pH →H, j:qH →H the inclusion maps.

a)We have i∈ L_E(pH, H). Withi^∗:H →pH,ξ 7→pξ, we have ii^∗=p and i^∗i= 1_H.

b) For every u ∈ qF p and v ∈ pF q put u¯ := j^∗ui, ¯v := i^∗vj. For u∈qF p, v∈pF q, and ξ∈pH we have

¯

u∈ LE(pH, qH), u¯^∗=u^∗, uξ¯ =uξ, uv = ¯u¯v.

c) The map

qF p→ LE(pH, qH), u7→u¯

is an isometry of Banach spaces. If p = q then it even is an isomrphism of C^∗-algebras.

d) Foru∈qF pwe have u^∗u=p iffhuξ¯ |uη¯ i=hξ|ηi for allξ, η∈pH.

In this case, pq.

e) For u∈qF p, u¯ is an isomorphism of Hilbert right E-modules iff u^∗u=p and uu^∗=q.

f) The statements below are equivalent.

f₁) pq.

f2) There is an r∈PrF such that qH ≈(pH)| (rH).

f₃) pH qH.

g) IfE is aW^∗-algebra andH is selfdual, then the statements below are equivalent.

g₁) pH ≈qH.

g₂) pH qH, qH pH.

Proof. a) For (ξ, η) ∈ pH×H we have hiξ|ηi = hξ|ηi =hpξ|ηi = hξ|pη i,so thati∈ LE(pH, H),ii^∗ =p and i^∗i= 1_H.

b) By a),

¯

u∈ L_E(pH, qH), u¯^∗ =i^∗u^∗j=u^∗,

¯

uξ=j^∗uiξ=j^∗uξ=j^∗quξ =quξ =uξ,

¯

u¯v=j^∗uii^∗vj =j^∗upvj=j^∗uvj =uv.

c) For u∈qF p and ξ∈H, we have uξ=upξ, so that by b), k¯uk= sup

ξ∈(pH)^#

kuξk= sup

ξ∈H^#

kuξk=kuk, i.e., the map preserves the norms.

Letw∈ LE(pH, qH). Putu:=jwi^∗.By a),u∈qF pand ¯u=j^∗jwi^∗i= w,so that the map is surjective. The last assertion follows from b).

d) By b),

huξ¯ |uη¯ i=huξ|uη i=hu^∗uξ|ηi.

If u^∗u = p then huξ¯ |uη¯ i = hξ|ηi. If huξ¯ |uη¯ i = hξ|ηi for all ξ, η ∈pH, then

hu^∗uζ|ηi=hu^∗upζ|ηi=hpζ|ηi, u^∗uζ=pζ, u^∗u=p forζ ∈H. The last assertion follows fromp∼uu^∗=quu^∗ ≤q.

e) follows from d).

f) f₁)⇒f₂) There isr ∈PrF withp∼r≤q.By e), qH ≃rH| (q−r)H =pH| (q−r)H.

f₂)⇒f₃) is trivial.

f₃) ⇒ f₁) There is a Hilbert right E-module K and an isomorphism ϕ:qH →(pH)| K of Hilbert right E-modules. Put

ψ: (pH)| K →pH, (ξ, η)7→η.

Then

ψ^∗:pH →(pH)| K, ξ7→(ξ,0), ψ^∗ψ: (pH)| K→(pH)| K, (ξ, η)7→(ξ,0),

ψψ^∗ :pH →pH, ξ7→ξ.

By c), there isu∈qF pwith ¯u=ϕ⁻¹◦ψ^∗.By b) and ([C1], Corollary 5.6.1.12)), u^∗u=u^∗u¯= ¯u^∗u¯=ψ◦ϕ◦ϕ⁻¹◦ψ^∗ =ψ◦ψ^∗,

uu^∗= ¯uu^∗ = ¯uu¯^∗ =ϕ⁻¹◦ψ^∗◦ψ◦ϕ≤ϕ⁻¹◦ϕ, so that, by c),

u^∗u=p, uu^∗ ≤q, pq.

g) By ([C1], Theorem 5.6.3.5 b)), F is a W^∗-algebra. By f₃) ⇒ f₁) we have pq and q p,so that p∼q and the assertion follows from e).

Lemma 2.5. Let E be aC^∗-algebra, H a Hilbert right E-module, andH the set of complementable submodules of H ordered by the inclusion relation.

a) The map

PrLE(H)→H, p7→pH := Imp is an isomorphism of ordered sets.

b) Forp, q∈ LE(H),

p∼q ⇔pH ≈qH, pq ⇔pHqH.

Proof. a) By Lemma 2.2 a), pH ∈H for every p∈ LE(H). The map is surjective since every element of His complemetable. Let p, q∈PrL_E(H). If p ≤ q then qξ = qpξ = pξ = ξ for ξ ∈ pH, so that pH ⊂ qH and the map preserves the order. Conversely, ifpH ⊂qH thenqpξ =pξ forξ ∈H, so that qp=p and p≤q. Thus the map is an isomorphism of ordered sets.

b) By Proposition 2.4 e), p ∼ q ⇔ pH ≈ qH. If p q then there is r ∈ L_E(H) withp∼r≤q. By the above and a),

pH≈rH ⊂qH, qH =rH| qH∩(rH)^⊥

, pH qH.

IfpH qH then there is a Hilbert rightE-moduleKsuch thatqH ≈K| pH.

Using this isomorphism we obtain an L ∈ H with pH ≈ L ⊂ qH. By a), pq.

Lemma 2.6. If E is a C^∗-algebra, then the statements below are equiva- lent for all Hilbert right E-modules H and K.

a) There is a Hilbert right E-moduleL such that K≈H| L.

b) There is a surjective u∈ L_E(K, H).

c) There is an injective v∈ LE(H, K) with closed range.

Proof. a)⇒ b) and a)⇒ c) are obvious.

b)⇒a). Sinceuis surjective,u^∗is injective. By ([C1], Corollary 5.6.1.12)), u^∗u∈ LE(K)+ and put |u|:= (u^∗u)^1/2 ∈ LE(K)+.Let ξ ∈ K with|u|ξ = 0.

Then u^∗uξ=|u|²ξ = 0 anduξ= 0, since u^∗ is injective. Thus, the map w: Im|u| →H, |u|ξ 7→uξ

is well-defined. For ξ, η ∈ K we have h |u|ξ| |u|ηi = hu^∗uξ|ηi =huξ|uηi, i.e.,wpreserves the scalar products. Being surjective, Im|u|is a closed Hilbert E-submodule of K and Im|u| ≈H. By ([W], Corollary 15.3.9)), Im|u| is complementable and we get

H= Im|u| | (Im|u|)^⊥≈H| (Im|u|)^⊥.

c) ⇒a). By ([W], Corollary 15.3.9), Imv is complementable, i.e., K= Imv| (Imv)^⊥.

By b) ⇒ a),H ≈Imv, so thatK ≈H| (Imv)^⊥.

Proposition 2.7. If E is a W^∗-algebra then the statements below are equivalent for all selfdual Hilbert right E-modules H andK.

a) H≈K.

b) HK and K H.

c) LE(H, K) has a bijective map.

d) LE(H, K)has a surjective map and an injective map with close range.

Proof. a)⇒ b) and a)⇒ c)⇒ d) are trivial.

d) ⇒ a) follows from Lemma 2.6, b) ⇒ a), c)⇒ a).

b) ⇒ a). Put L:=H| K.By Lemma 2.5 a), there arep, q∈PrL_E(L) with H=pL,K=qL.By Lemma 2.5 b), we get successively

pq, qp, p∼q, H≈K.

Remark. The example

K→K, ξ 7→2ξ

shows that a bijective map ofLE(H, K) is not necessarily an isomorphism.

Corollary 2.8. If E is a W^∗-algebra then the statements below are equivalent for all Hilbert right E-modules H, K, L.

a) K ≈H| L.

b) There is an exact sequence

0→H→K →L→0 with adjointable homomorphisms.

Proof. (a)⇒ (b) is trivial.

(b) ⇒ (a). Let u∈ LE(H, K), v ∈ LE(K, L) be the homomorphisms of b). It follows from Imu = Kerv that u has closed range. The restriction of v to (Imu)^⊥ = (Kerv)^⊥ is bijective and so, by Proposition 2.7, c) ⇒ a), we have

H ≈Imu, L≈(Imu)^⊥. Thus,K ≈Imu| (Imu)^⊥≈H| L.

Lemma 2.9. LetE be aC^∗-algebra,r ∈PrE,F :=rEr, andϕ:F →E the inclusion map.

a) If p∈PrE,q ∈PrF, and pϕq, then there isq^′ ∈PrF with q^′ ≤q, p∼ϕq^′.

b) If p, q∈PrF then

p∼q⇔ϕp∼ϕq, pq ⇔ϕpϕq.

c) If E is a W^∗-algebra then c1) ϕ defines an injective map

ψ: (PrF/∼) → (PrE/∼);

c₂)ψ(PrF/∼)is a hereditary set of(PrE/∼), to mean thatq ∈(PrE/∼) belongs to ψ(PrF/∼) if there is p∈(PrF/∼) withq ψ(p);

c3) the map (PrF/∼)→ ψ(PrF/∼) defined by ψ is an isomorphism of ordered sets;

c4) if (PrE/∼) is order complete, then so is (PrF/∼), and ψ comutes with the suprema and infima.

Proof. a) By the hypothesis, there isp^′ ∈PrEwithp∼p^′ ≤ϕq.SinceF is a hereditaryC^∗-subalgebra ofE ([C1], Example 4.3.4.2)), there isq^′∈PrF with q^′ ≤q, ϕq^′ =p^′ ∼p.

b) Assume ϕpϕq. Then there is u ∈E with u^∗u=ϕp,u u^∗ ≤ϕq. It follows that

u=u ϕp∈Er, u=u u^∗u= (ϕq)u u^∗u= (ϕq)u∈rE, u∈rEr=F, pq.

The implication

ϕp∼ϕq⇒p∼q is proved similarly. The other implications are trivial.

c) follows from a) and b). More precisely, c₁) follows from b).

c₂) follows from a).

c₃) follows from b) and c₂).

c4) follows from c2) and c3).

Definition 2.10. For every W^∗-algebra E we denote by H(E) the class of all selfdual Hilbert right E-modules (with identification of the isomorphic ones) endowed with the associative and commutative addition | and the order relation(Proposition 2.7, b)⇒a)). {0}is the neutral element for the addition and the smallest element with respect to the order relation.

Theorem 2.11. Let E be a W^∗-algebra and for every setI consider the selfdual Hilbert right E-module

E⊗l²(I)≈ ^W|

i∈I

E (Theorem1.4 a)) and the W^∗-algebra

E_I :=E⊗ L(l²(I))≈ L_E(E)⊗ L(l²(I))≈ L_E⊗¯K(E⊗l²(I))≈ L_{E W}

|

i∈I

E

([C2], Theorem2.4 d)).

a) ForI ⊂J the natural imbedding ψ_IJ : ^W|

i∈I

E → ^W|

j∈J

E

produces the map

ϕ_IJ :E_I→E_J, u7→ψ_IJu ψ_IJ^∗ .

ϕ_IJ is an injective W^∗-homomorphism and its image isr_IJE_Jr_IJ, where rIJ :=

E¨J

X

i∈I

1E⊗ h · |eiiei.

b) (PrE_I)_I and the associated maps (ϕ_IJ)_IJ form an inductive system.

For p ∈ PrE_I, q ∈ PrE_J write p ∼ q if there is a set K ⊃ I ∪ J with ϕIKp∼ϕJKq. ThenϕILp∼ϕJLq for every L⊃I∪J.

Denote by PR(E)the class of S

IPrE_Iwith the identification of the ele- mentsp∼q;PR(E)is in some sense the inductive limit of the above inductive system.

c) For p ∈ PrE_I, q ∈ PrE_J, write p q if there is K ⊃ I ∪J with ϕ_IKpϕ_JKq. Then ϕ_ILpϕ_JLq for every L⊃I∪J and is an order relation on PR(E).

d) For every p∈PrEI we have Imp=p

_W |

i∈I

E

∈ H(E) and, for p∈PrE_I and q∈PrE_J,

p∼q⇔Imp≈Imq, pq⇔ImpImq while the deduced map

PR(E)→ H(E), p7→Imp

is bijective and we may identify PR(E) and H(E) as ordered classes.

e) If PR(E) is order complete (equivalently: H(E) is order complete) then (PrE/∼) is order complete and may be identified as ordered set with a hereditary subset of PR(E).

f) Fix a set I0. For every I ⊃I0 we have

E_I=E⊗ L(l²(I))≈E⊗ L(l²(I₀))⊗ L(l²(I\I₀))≈E_I0⊗ L(l²(I\I₀)).

By this relation, PR(E) and PR(EI0) may be identified.

Proof. a) We have

ψ_IJ^∗ ψ_IJ = 1_l²_(I), ψ_IJψ_IJ^∗ =r_IJ, ψ_IJ^∗ r_IJ =ψ^∗_IJ, r_IJψ_IJ =ψ_IJ and, for u∈E_I,

ψ_IJ^∗ (ϕ_IJu)ψ_IJ =ψ^∗_IJψ_IJu ψ_IJ^∗ ψ_IJ =u, r_IJ(ϕ_IJu)r_IJ =r_IJψ_IJu ψ_IJ^∗ r_IJ =ψ_IJu ψ_IJ^∗ ,

i.e., Imϕ_IJ ⊂r_IJE_Jr^∗_IJ.Ifv∈r_IJE_Jr_IJ thenψ^∗_IJv ψ_IJ ∈E_I and ϕIJ(ψ_IJ^∗ v ψIJ) =ψIJψ^∗_IJv ψIJψ_IJ^∗ =rIJv rIJ =v, i.e., Imϕ_IJ=r_IJE_Jr_IJ.Moreover, foru∈E_Iand (a, ξ, η)∈E¨×^W|

j∈J

E×^W|

j∈J

E we have

D

ϕ_IJu,(a, ξ, η)^ E

=h h(ϕ_IJu)ξ|ηi, ai=h hψ_IJu ψ_IJ^∗ ξ|ηi, ai=

=h hu ψ_IJ^∗ ξ|ψ_IJ^∗ ηi, ai=D u,

z ^}| { (a, ψ^∗_IJξ, ψ^∗_IJη)E

, i.e.,

ϕ^′_IJ(a, ξ, η) =^

z ^}| { (a, ψ^∗_IJξ, ψ^∗_IJη)

and ϕ_IJ is a W^∗-homomorphism ([C1], Theorem 5.6.3.5 b) and Proposition 4.4.4.6)).

b) It follows from a) (the last assertion of b) from the last assertion of a)).

c) It follows from a) and b).

d) By Lemma 2.2, Imp ∈ H(E). The relations follow from Proposi- tion 2.4 c), e), f), g). By Proposition 2.1 and ([C1], Theorem 5.6.4.10 a)), the map is surjective.

e) It follows from a) and Lemma 2.9 c).

f) This is obvious.

3. SOME TECHNICAL RESULTS

We start with

Lemma 3.1. Let E be a W^∗-algebra, α an ordinal number, (p_γ)_γ∈α a family in PrE, β the smallest element of α with W

β∈γ∈αp_γ 6≤ p_β, F :=

E⊗ L(l²(α)), and ω_γ := h · |e_γ ie_γ ∈ PrL(l²(α)) for every γ ∈ α. Then there is a family (q_γ)_γ∈α such that

1. γ∈β ⇒qγ =pγ, 2. p_β ≤q_β,

3. β∈γ ∈α⇒q_γ≤p_γ, 4. γ≤β ⇒W

γ∈δ∈αq_δ≤q_γ, 5.(q_β−p_β)⊗ω_β ∼

F¨

P

γ∈α\{β}

(p_γ−q_γ)⊗ω_γ, 6.

F¨

P

γ∈α

p_γ⊗ω_γ ∼

F¨

P

γ∈α

q_γ⊗ω_γ.