Math. Proc. Camb. Phil. Soc. (1999), 127, 295

Printed in the United Kingdom c© 1999 Cambridge Philosophical Society

295

Geometry of the sphere of a Hilbert module

By ESTEBAN ANDRUCHOW,

Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.A. Roca 850,

1663 San Miguel, Argentina.

GUSTAVO CORACH and DEMETRIO STOJANOFF

Instituto Argentino de Matematica, Saavedra 15,

1083 Buenos Aires, Argentina.

and

Departamento de Matematica, FCEN, Universidad de Buenos Aires,

Ciudad Universitaria, 1428 Buenos Aires, Argentina.

(Received 23 March, 1998)

Abstract

The sphere SX = x ∈ X: 〈x, x〉 = 1 of a right Hilbert C∗-module X over a unitalC∗-algebra B is studied using differential geometric techniques. An action of theunitary group of the algebra LB(X) of adjointable B-module operators of X makesSX a homogeneous space of this group. A reductive structure is introduced, as wellas a Finsler metric. Metric properties of the geodesic curves are established. In thecase B a von Neumann algebra and X self-dual, the fundamental group of SX iscomputed.

1. Introduction

The theory of Hilbert C∗-modules was initiated by Kaplansky [15], for commuta-tive C∗-algebras, and by Paschke [23] and Rieffel [28] for non-commutative operatoralgebras. The subject got a great impulse with the work of Kasparov [16] on the foun-dations of KK-theory and is now a very useful tool in many areas such as inducedrepresentations, K-theory of operator algebras, quantum groups, and so on.

Hilbert C∗-modules come up naturally when studying the differentiable geometryof (infinite dimensional) homogeneous spaces over the group G of invertibles of a C∗-algebra A. Suppose that the isotropy group G0 of the homogeneous space K overG is the group of invertibles of a C∗-subalgebra B of A. As in classical differentialgeometry, one looks for a reductive structure for K ' G/G0, i.e. a G0-invariantsupplement H for the Lie algebra G0 of G0 in the Lie algebra G of G. In our case,one gets such a supplement by finding a faithful conditional expectation E : A→ B,namely H = ker E. In the general case, this space H is a model for the tangentspaces of K. If H = ker E, then H has a natural pre-Hilbert C∗-module structure(which is complete if the index of E is finite [6, 26]), i.e. the tangent spaces of

296 E. Andruchow, G. Corach and D. StojanoffK become also pre-Hilbert modules. Along with the inner product, this structurecarries also a norm, which gives rise to an invariant Finsler metric for K. We believethat Hilbert C∗-modules will prove a useful tool in infinite dimensional differentialgeometry. A preliminary study along these lines can be found in [2].

Let B be a unital C∗-algebra and X a right Hilbert module over B, we shall beconcerned with the unit sphere of X, i.e.

SX = x ∈ X: 〈x, x〉 = 1.The object of this paper is to carry on a topological and differential geometric studyof this set.

The unitary group ULB (X) of the C∗-algebraLB(X) of adjointableB-module oper-ators of X acts naturally on SX : U • x = U (x), for U unitary and x ∈ SX . ThereforeSX , or rather its connected components, can be presented as quotient spaces ofULB (X). Namely, if x0 ∈ SX , the connected component of x0 in SX can be regardedas the connected component of the coset of the identity in the quotient ULB (X)/Ix0 ,where Ix0 denotes the isotropy group of the action at x0, i.e.

Ix0 = V ∈ ULB (X) : V (x0) = x0.In Section 2 it is shown that this quotient admits smooth cross sections and there-

fore SX becomes a homogeneous C∞ space, as well as a complemented submanifoldof X. Also some elementary properties of the action are shown.

In Section 3 a reductive structure for SX is introduced. As in classical differentialgeometry, a reductive structure for a homogeneous space G/K is a decompositionG = K ⊕H (G,K are the Lie algebras of G,K) such that the supplement H isinvariant under the action of K. This construction is carried on explicitly, and usingstandard techniques ([2, 22]) the geodesics of the linear conection induced in SX bythis reductive structure are computed. Also a natural Finsler metric is defined.

In Section 4 we study some properties of this Finsler metric. Once a Finsler metricis defined, the main problem is to know whether the geodesics of the connectionare minimal or locally minimal for this metric. Minimality of geodesics (in the in-finite dimensional setting) has only been established basically in few cases: unitaryoperators [4], self-adjoint invertible operators [9], self-adjoint projections [27] andpositive part of square roots of invertible elements with fixed unitary part [20]. Weuse the results of [27] to obtain local minimality of certain geodesics with specialdirections.

In Section 5 we study the case when B is a von Neumann algebra and the moduleX is self-dual (see [23]). Certain facts concerning the type decomposition of B andLB(X) are considered. Particular interest is paid to the case in which B and LB(X)are finite. In this case, a possibly unbounded operator D affiliated to the centre of Bis defined, which is a sort of index for the module X. If X is a von Neumann algebraA and the inner product is given by a finite index conditional expectation E : A→ B,then the operator D is bounded and equals E([E]), where [E] is the index operatorfor E [6].

The main goal of Section 6 is the computation of the fundamental group of SX . Bythe results of [11], it suffices to consider the finite case, because infinite von Neumannalgebras have unitary groups with trivial fundamental groups. The index operatorD of X is useful in the computation of the fundamental group of the spheres SX .

Geometry of the sphere of a Hilbert module 297Typically, lower case letters x, y, z will denote elements of X while upper case

letters T, Y, Z, U, V will denote operators in LB(X).The standard references concerning Hilbert modules will be the papers [23, 24, 28],

and the excellent book [18].

2. Modular spheres

Let B be a unital C∗-algebra and X a Banach space which is also a pre-HilbertrightB-module. Denote by 〈 , 〉 theB-valued inner product. Later on we will supposethat X is complete with the Hilbert module norm ‖x‖ = 〈x, x〉 1

2 . However, the firstresult does not require completeness in the norm ‖ ‖. Suppose that 〈 , 〉 is continuous;then it is clear that the set X−1 = x ∈ X: 〈x, x〉 ∈ GB is an open subset of X.

Proposition 2·1. The sphere SX = x ∈ X: 〈x, x〉 = 1 is a C∞-submanifold of X.

Proof. Indeed, S is a submanifold of X−1 since the map

X−1 → GB wB+, x 7→ 〈x, x〉is aC∞ submersion. This is clear because the map has smooth cross sections. Namely,if a = 〈x0, x0〉 ∈ GB then

GB wB+ → X−1, b 7→ x0 · (a− 12 b

12 )

is such a map.

An example of the above situation is given when X is a unital C∗-algebra A, Bis a C∗-subalgebra of A (with the same unit) and there exists a faithful conditionalexpectation E : A→ B. The usual left Hilbert module structure is given by 〈a1, a2〉 =E(a∗1a2).

From now on, X is assumed to be complete. In the particular case X = A thisoccurs if and only if the conditional expectation E has finite index ([6, 10]).

If A is a C∗-algebra, denote by G = GA the group of invertibles of A, U = UA theunitary group of A and P(A) = q ∈ A : q2 = q∗ = q. If p ∈ P(A), put

Vp(A) = vp ∈ Ap : pvv∗p = pand

Ep = q ∈ P(A) : there exists v ∈ A with v∗v = p, vv∗ = q.In the C∗-algebra language, Vp(A) consists of all the partial isometries of A withinitial space p and Ep of all projections of A which are equivalent to p. Both sets areC∞ manifolds, Vp(A) is studied in [1] and there are many papers considering thegeometric structure of the space of projections of a C∗-algebra (see [7, 8, 27, 32]).In [1] it is shown that Ep consists of a union of connected components of the spaceP(A).

Let LB(X) denote the usual C∗-algebra of B-linear adjointable operators actingon X. The manifold SX is invariant under the action of the unitary group of LB(X).If x, y ∈ X, denote by x⊗ y the element of LB(X) given by

x⊗ y(z) = x〈y, z〉.Note that if x ∈ SX , then x ⊗ x is a self-adjoint projection. Let us fix the elementx0 ∈ SX . Then for every other x ∈ SX , the element v = x⊗ x0 is a partial isometry

298 E. Andruchow, G. Corach and D. Stojanoffin LB(X) with initial space e = (x ⊗ x0)∗(x ⊗ x0) = x0 ⊗ x0 and final space x ⊗ x.Note that in particular all projections of the form x⊗x are equivalent (to e). Let usconsider the C∞ map

% : SX → Ee, %(x) = x⊗ xand the linear operator

x0 : X →LB(X)e, x0 (x) = x⊗ x0.

Note that x0 |SX takes values in Ve(LB(X)).Observe that if w ∈ X and x ∈ SX , then ‖w‖ = ‖w ⊗ x‖.Proposition 2·2. The operator x0 is an isometric surjective isomorphism between the

Banach spaces X and LB(X)e, which maps SX isometrically onto Ve(LB(X)).

Proof. If T ∈LB(X), then Te = T (x0 ⊗ x0) = T (x0)⊗ x0 = x0 (T (x0)).

‖x0 (x)‖ = ‖x⊗ x0‖ = ‖x‖.q

The aim of this paper is to study the geometry of SX . The action of the unitarygroup ULB (X) plays a major role in this study. First, let us prove that the action ofULB (X) is locally transitive in SX .

Proposition 2·3. Let x1, x2 ∈ SX . There exists a unitary element U in ULB (X) withU (x1) = x2 if and only if the final projections x1⊗x1 and x2⊗x2 are unitary equivalentin LB(X).

Proof. If there exists U ∈ ULB (X) with U (x1) = x2 then x2 ⊗ x2 = U (x1)⊗U (x1) =U (x1 ⊗ x1)U∗.

On the other hand, suppose that there exists U ∈ ULB (X) with U (x1 ⊗ x1)U∗ =x2⊗x2. ThenW1 = U (x1⊗x0) andW2 = x2⊗x0 are partial isometries with initial spacee and final space f = x2⊗x2. Then W = W1W

∗2 +1−f is a unitary element of LB(X)

which verifiesW (x2⊗x0) = W1W∗2 W2 = U (x1⊗x0). Therefore x2⊗x0 = W ∗U (x1⊗x0)

and this implies that x2 = W ∗U (x1).

Proposition 2·4. If x1, x2 ∈ SX satisfy that ‖x1−x2‖ < 12 , then there exists a unitary

element U of LB(X) such that U (x1) = x2.

Proof. The statement follows from the inequalities

‖x1 ⊗ x1 − x2 ⊗ x2‖ 6 ‖x1 ⊗ x1 − x1 ⊗ x2‖ + ‖x1 ⊗ x2 − x2 ⊗ x2‖= ‖x1 ⊗ (x1 − x2)‖ + ‖(x2 − x1)⊗ x2‖ 6 2‖x1 − x2‖,

using the well known fact (see for example [8]) that projections that lie at distanceless than 1 are unitarily equivalent.

Fix x0 ∈ SX and consider the C∞ map

Πx0: ULB (X) → SX , Πx0 (U ) = U (x0).

We will show that these mappings are submersions and define a homogeneousstructure on SX . In order to do so it suffices to construct C∞ local cross sectionsfor Πx0 , x0 ∈ SX . The way the unitary element linking two given elements of SX is

Geometry of the sphere of a Hilbert module 299constructed in Proposition 2·4, enables one to compute explicit cross sections for themaps Πx0 .

Proposition 2·5. For any x0 ∈ SX the map

Πx0: ULB (X) → SX

has C∞ local cross sections. Namely, given x ∈ SX with ‖x0 − x‖ < 12 then there exists

µx0 (x) ∈ ULB (X) such that Πx0 µx0 (x) = x for all such x (i.e. µx0 (x)(x0) = x) and themap

µx0: x ∈ SX: ‖x− x0‖ < 12 → ULB (X)

is C∞.

Proof. As shown before, the projections x⊗ x and x0 ⊗ x0 lie at distance less than1. Therefore T = (x⊗x)(x0⊗x0) + (1−x⊗x)(1−x0⊗x0) is invertible in LB(X) andsatisfies T (x0⊗x0) = (x⊗x)T . Then the unitary part of T in its polar decompositionT = V |T | satisfies also V (x0 ⊗ x0) = (x ⊗ x)V . Note that V = T (T ∗T )−

12 . Then, as

shown in the proposition above, x⊗x0 +V (1−x0⊗x0) is a unitary element of LB(X)linking x0 with x. Replacing the explicit value of V , straightforward computationsshow that

µx0 (x) = x⊗ x0 + V (1− x0 ⊗ x0)

is a unitary element of LB(X) and Πx0 (µx0 (x)) = x for all x ∈ SX with ‖x−x0‖ < 12 .

Corollary 2·6. SX is a C∞ homogeneous space under the action of the Banach–Lie group ULB (X). The isotropy groups of the action are the subgroups Ix0 ⊂ ULB (X),Ix0 = V ∈ ULB (X): V (x0) = x0.

Remark 2·7. The cross section defined in Proposition 2·5 takes values in thesubgroup of unitaries of I + KB(X) = λ1 + T :λ ∈ C, T ∈KB(X), where KB(X)denotes the two sided ideal of compact operators in LB(X), i.e. the norm closure ofthe linear span of the operators of the form x ⊗ y, x, y ∈ X. Therefore one couldregard SX as a homogeneous space of this smaller group. In some cases both groupscoincide ([10, 31]).

Our next step will be to introduce a reductive structure for SX (which in turn willinduce a linear connection in the manifold SX). This will be done in the next section.The remaining of this is devoted to the following example.

Example 2·8. Let B ⊂ A be unital C∗-algebras (having the same unit) andE:A → B a faithful conditional expectation. Consider A as a B-right module withthe usual product and put 〈a, a′〉E = E(a∗a′). Then A is a pre-Hilbert module. Letus recall the Jones generalized basic construction and some elementary facts aboutindex for conditional expectations (see for example [6] or [12]). There is a faithfulrepresentation of A in a Hilbert space H and a self-adjoint projection pE (calledJones projection) in L(H) with the following properties (we shall identify A with itsimage under this representation).

Denote by A1 the C∗-subalgebra of L(H) generated by A and pE. Then:

300 E. Andruchow, G. Corach and D. Stojanoff(i) pEapE = E(a)pE, a ∈ A;

(ii) pE′ wA = B;(iii) pEA1pE is isomorphic to B via the *-isomorphism x 7→ xpE = pExpE.

For the details concerning this construction see for example [2] or [6].

Recall ([6, 10, 26]) that E is said to have finite index if there exists a positive realnumber κ such that κE − IdA is a positive linear map on A. The following result iswell known (see [2] or [10]).

Theorem 2·9. The following conditions are equivalent:(i) E has finite index;

(ii) There exists λ > 0 such that λE − IdA is completely positive;(iii) A is complete with the Hilbert module norm given by E;(iv) ApE is closed in norm in A1;(v) ApE = A1pE.

Suppose that E has finite index. Then A is a Hilbert module and therefore LB(A)is a C∗-algebra. Note that E ∈LB(A) is a self-adjoint projection. There is a natural*-isomorphism (not necessarily onto)

Φ: A1 →LB(A)

such that A 3 a 7→ La and pE 7→ E, where La(x) = ax for x ∈ A.The following facts are known:

Theorem 2·10.(a) If A and B are von Neumann algebras and E has finite index and is normal

then Φ is onto, i.e. LB(A) is generated as a von Neumann algebra by pE andA. Moreover, the finite index condition implies that A is not only complete but aselfdual Hilbert module as well ([6]).

(b) In the C∗-algebra case, A1 coincides with LB(A) if and only if E satisfies astronger index condition. Namely, if E has strongly finite index, a fact equivalentto saying that A is a finitely generated projective B-module ([31]).

The object of our study is, then, the set SA = x ∈ A: E(x∗x) = 1. It followsfrom the elementary properties of the basic construction that elements of the forma0 +

∑i aipEa

′i, with a0, ai ∈ A are norm-dense in A1.

Since the index of E is finite, ApE = A1pE. This equality implies that the map

A→ A1pE , a 7→ apE

is a Banach space isomorphism. Let us denote by θ its inverse. In other words, ifx ∈ A1, θ(x) is defined by the relation

θ(x) ∈ A and xpE = θ(x)pE .

Thus, via the isomorphism Φ, θ is identified with the inverse of the map x, forx = 1 ∈ SA.

If z ∈ A1 we can describe the element Φ(z) ∈ LB(A): Φ(z)(a)pE = zapE andtherefore Φ(z)(a) = θ(zapE).

It has been shown that the action of ULB (X) on SX is locally transitive. In somecases more can be said:

Geometry of the sphere of a Hilbert module 301Corollary 2·11. If A is a finite von Neumann algebra and E:A → B is a finite

index conditional expectation then the sphere SA is connected and the action of ULB (A)

(= UA1 ) is transitive.

Proof. Indeed, since A is finite, all final projections corresponding to elements ofSA are unitarily equivalent and therefore applying Proposition 2·3 the conclusionfollows.

Trivial examples show that in many cases SA can be disconnected, even whenthe unitary group of A is connected. Take for instance A = B = L(l2(N)) and E =Id. Then different integer powers of the unilateral shift lie in different connectedcomponents of SA = a ∈ L(l2(N)) : a∗a = 1.

3. Reductive structure

Fix x0 ∈ SX and put e = x0 ⊗ x0. Then the isotropy group

Ix0 = V ∈ ULB (X) : V (x0) = x0,consists of the unitaries V such that V e = e. Using the matrix decomposition ofLB(X) induced by the projection e

V =(

1 00 V2,2

).

The Lie algebra Ix0 of Ix0 consists of matrices of the form

Z =(

0 00 Z2,2

),

with Z∗2,2 = −Z2,2. We are looking for a supplement of I1 in the Lie algebra of ULB (X)

(identified with LB(X)ah = Z ∈ LB(X) : Z∗ = −Z). The most natural choice isthe space Hx0 of skew-hermitian matrices of the form

W =(

W1,1 W1,2

−W ∗1,2 0

),

with W ∗1,1 = −W1,1. Hx0 = W ∈LB(X)ah: (1− e)W (1− e) = 0.Thus, we have defined a smooth distribution SX 3 x 7→Hx satisfying VHxV

∗ = Hx

for any V ∈ Ix and Hx ⊕ Ix = LB(X)ah, where smooth means that if Px is the(real) linear projection onto Hx given by the decomposition Hx ⊕Ix = LB(X)ah,then the mapping x 7→ Px ∈ L(LB(X)ah) is C∞.

Such an object is what in classical differential geometry is called a reductive struc-ture. We shall not go into the details, but will describe the main invariants inducedby this reductive structure. For general facts about this object the reader is referredto [17], or [22] where the infinite dimensional case is particularly considered.

As it is usual with reductive spaces, a linear connection is induced in SX . In orderto compute its invariants it will be useful to describe the isomorphism

κx0: T (SX)x0 →Hx0

which is the inverse of the derivative

d(Πx0 )1|Hx0: Hx0 → T (SX)x0 .

302 E. Andruchow, G. Corach and D. StojanoffNote that d(Πx0 )1(Z) = Z(x0).A typical tangent vector v in T (SX)x0 is an element v ∈ X such that

〈v, x0〉 + 〈x0, v〉 = 0.

We want to compute κx0 (v), which should be a matrix (using e = x0⊗x0) in LB(X)ah

of the form Z =(

Z1,1 Z1,2

−Z∗2,1 0

)with Z(x0) = v. These conditions are fulfilled by

the (unique) element

κx0 (v) = Z = v ⊗ x0 − (x0 ⊗ v)(1− e).In order to verify that this element satisfies the required conditions, first note that

the equality 〈v, x0〉 + 〈x0, v〉 = 0 implies that (x0 ⊗ v)e = −e(v ⊗ x0). It is also clearthat e(x0 ⊗ y) = x0 ⊗ y and (y ⊗ x0)e = y ⊗ x0 for any y ∈ X.

Therefore

Z∗ = x0 ⊗ v − (1− e)(v ⊗ x0) = e(x0 ⊗ v)− x0 ⊗ v − e(v ⊗ x0) = −Z.Moreover, Z ∈ Hx0 , i.e. (1 − e)Z(1 − e) = 0. Finally, Z(x0) = v〈x0, x0〉 = v, because(1− e)(x0) = 0.

If γ is a smooth curve in SX with γ(0) = x, then the solution Γ of the lineardifferential equation

Γ(t) = κγ(t)(γ(t))Γ(t)Γ(0) = 1

is a smooth curve Γ in ULB (X) satisfying Πx(Γ) = γ and Γ ∈ Hγ . This is usuallycalled the transport equation, since Γ acting on vectors of T (SX)x yields the paralleltransport along γ of the induced linear connection. The covariant derivative can becomputed as follows: if v is a tangent vector field along γ ∈ SX , then

D

dtv = Γ

(d

dt[Γ∗(v)]

),

where Γ is the solution of the transport equation for γ with x = γ(0). In our case thecovariant derivative is given by

D

dtv = v(t)− γ(t)〈γ(t), v(t)〉 + γ(t)〈γ(t), v(t)〉 + γ(t)〈γ(t), γ(t)〉〈γ(t), v(t)〉.

The geodesics of this connection can be computed in a standard fashion ([22]).Namely, the unique geodesic δ with δ(0) = x and δ(0) = v is given by

δ(t) = etκx(v)(x).

The next step is the definition of a Finsler metric on SX . If x ∈ SX and v ∈ T (SX)x,put

‖v‖x = ‖v‖ = ‖< v, v > ‖ 12 .

Recall that if w ∈ X and x ∈ SX , then ‖w‖ = ‖w ⊗ x‖.Let γ : [0, 1]→ SX be a smooth curve with origin x0. Then the length of γ is given

by∫ 1

0 ‖γ(t)‖γ(t)dt.If Γt is the solution of the transport equation, then Γt(x0) = γ(t). Therefore

Geometry of the sphere of a Hilbert module 303‖Γt(x0)‖ = ‖Γt(x0)⊗ x0‖ = ‖Γte‖. Then∫ 1

0‖γ(t)‖γ(t)dt =

∫ 1

0‖Γte‖dt.

In other words, the length of γ equals the length of the curve given by the firstcolumns of the unitaries Γ (regarded as matrices in terms of e = x0 ⊗ x0), i.e. thelength of Γe in ULB (X)e.

We do not know if the geodesic curves are locally minimal with respect to thisFinsler metric just defined. However, we will show that the sphere SX can be locallydecomposed in each point into two ‘orthogonal’ parts, a copy in SX of the neigh-bourhood q ∈ Ee: ‖q − e‖ < 1 ⊂ P(LB(X)), and a ‘normal’ part. It will be shownthat these two submanifolds of SX have minimality properties for certain geodesics.

4. Metric properties

For x0 ∈ SX fixed, put e = x0 ⊗ x0 as before, and let Cx0X be the subset of SX

Cx0X = eZ(x0) : with Z∗ = −Z, eZ = Z(1− e) and ‖Z‖ < π/2.

Proposition 4·1. Cx0X is a C∞ submanifold of SX and the C∞ map

% : Cx0X → q ∈ Ee: ‖q − e‖ < 1, %(x) = x⊗ x

is a diffeomorphism.

Before the proof, let us state the following Lemma, whose proof resembles that ofan analogous result in [1].

Lemma 4·2. Let A be a C∗-algebra, p ∈ P(A) and

D = x ∈ A : x∗ = −x, px = x(1− p) and ‖x‖ < π/2.Then the mapping

exp: D→ exp: x ∈ D, exp(x) = exp

is bicontinuous.

Proof. Let x ∈ A with x∗ = −x and px = x(1 − p). Using the decomposition of Ainduced by p,

x =(

0 a−a∗ 0

)and

ex =(

cos (|a|) af (|a|)−a∗f (|a∗|) cos (|a∗|)

),

where f (t) = (sin t)/t (t > 0). Put ε = 1− 2p. Then condition px = x(1− p) becomesεx = −xε and therefore exε = εe−x.

The mapping exp is obviously onto. Let us prove that it is also ‘one to one’.Suppose that ex1p = ex2p for some x1, x2 ∈ D. Then ex1εe−x1 = ex2εe−x2 and thereforee2x1ε = e2x2ε. Since ε is invertible, this implies e2x1 = e2x2 . Since

e2xi =(

cos (|ai|) aif (|ai|)−a∗i f (|a∗i |) cos (|a∗i |)

),

304 E. Andruchow, G. Corach and D. Stojanoffwith ‖ai‖ < π, i = 1, 2 and the function cos is a diffeomorphism on the set of posi-tive elements of A with norm strictly less than π, then cos (|a1|) = cos (|a2|) implies|a1| = |a2|. Another functional calculus argument shows that both f (|ai|), i = 1, 2 areinvertible elements of A, and therefore a1 = a2. Moreover, this same sort of argumentshows that exp is bicontinuous, since its inverse can be explicitly computed.

Proof of Proposition 4·1. First we prove that

x0 (Cx0E ) = eZe : Z∗ = −Z, eZ = Z(1− e), ‖Z‖ < π/2

is a submanifold of LB(X)e. Let D = Z ∈ LB(X)ah: eZ = (1 − e)Z, ‖Z‖ < π/2then, from Lemma 4·2, the map

D→ eZe : Z ∈ D,given by z 7→ eze, is bicontinuous.

This map endows x0 (Cx0X ) with a C∞ manifold structure. In order to see that

x0 (Cx0X ) is a submanifold of LB(X) define

ϕ : D⊕ V ∈LB(X)ah: V = eV e →LB(X)e

by

ϕ(Z + V ) = eZeV e;

ϕ is a local diffeomorphism around the points Z + 0. In other words, the derivativeof ϕ at these points is an isomorphism. Note that

d(ϕ)Z+0(Z ′ + V ) = d(exp)Z(Z ′)e + eZV e.

d(exp)Z(Z ′)e is the derivative of the diffeomorphism Z 7→ eZe considered above andis therefore an isomorphism between Z ∈LB(X)ah: eZ = Z(1−e) and the tangentspace of x0 (C

x0X ) at Z, i.e. eZZ ′ ∈ LB(X)ah: eZ ′ = Z ′(1 − e). On the other hand,

since V = V e the other summand is an isomorphism from V ∈LB(X)ah: V = eV eonto eZV ∈LB(X)ah: V = eV e. These two spaces have nil intersection and theirdirect sum equals eZX ∈ LB(X)ah : (1 − e)X(1 − e) = 0 = eZHx0 . Our claim isthen proved and this local diffeomorphism gives the local charts that endow x0 (C

x0X )

the (complemented) submanifold structure. Since x0 is a linear isomorphism, thenCx0X is a submanifold of SX .The map % : Cx0

X → q ∈ Ee: ‖q − e‖ < 1, %(x) = x⊗ x is the composition of threediffeomorphisms, namely

x0: Cx0X ⊂ SX → eZe ∈LB(X)ah: Z ∈ D,

the inverse of the exponential

eZe ∈LB(X)ah: Z ∈ D → D

carrying eZe 7→ Z and the map

D→ q ∈ Ee: ‖q − e‖ < 1, Z 7→ eZee−Z

which is a diffeomorphism ([1]).

Proposition 4·3. For every x ∈ Cx0X there is a unique geodesic curve δ ⊂ Cx0

X withδ(0) = x0, δ(1) = x such that δ is minimal with respect to arc length among all the curvesin SX joining x0 and x.

Geometry of the sphere of a Hilbert module 305Proof. Since x ∈ Cx0

X , some Z ∈ D(= Z ∈ LB(X)ah: eZ = Z(1 − e), ‖Z‖ < π/2)exists such that x = eZ(x0). Then δ(t) = etZ(x0) is the required geodesic.

Let us see that it is minimal. Let γ(t) ∈ SX be another smooth curve with γ(0) = x0

and γ(1) = x. By means of the GNS construction one can choose a cyclic represen-tation (π,H, ξ) of LB(X) such that

π(eZ2e)ξ = −‖eZ2e‖ξ.Note that since Ze = (1 − e)Z, then ‖eZ2e‖ = ‖Z‖2. Also, it is apparent from thestatement above that π(e)ξ = ξ. Let Γ be the solution of the transport equation forγ. Recall that Γt is a smooth curve in ULB (X) with Γ0 = 1 and Γt(x0) = γ(t). Considerthe curves ∆(t) = π(etZe)ξ and Ω(t) = π(Γ(t)e)ξ in H.

Note that the curves ∆ and Ω lie in the unit sphere ofH, both joining ξ and π(eZ)ξ.The first fact is apparent because π(e)ξ = ξ. To prove the second assertion observethat Γe = Γ(x0⊗x0) = Γ(x0)⊗x0 = γ(t)⊗x0, and therefore Γ1e = γ(1)⊗x0 = x⊗x0 =eZ(x0)⊗ x0 = eZe.

∆ is a curve of length less than π/2. Indeed, since ∆(t) = π(etZZe)ξ, then

‖∆(t)‖ = ‖π(etZZe)ξ‖ = ‖π(Z)ξ‖ = ‖π(Ze)ξ‖= (− < π(eZ2e)ξ, ξ >)

12 = ‖Z‖ < π/2.

Moreover, this implies that the length of ∆ equals the length of δ. Let us prove that∆ is a geodesic of the Riemannian structure of the unit sphere SH of H.

The geodesic distance of any two vectors ζ, η in SH lying at distance less than πis given by dg(ζ, η) = arccos (Re < ζ, η >), where Re (λ) stands for the real part ofthe complex number λ. It is a standard fact that any smooth curve between thesetwo vectors whose length equals this distance must be a (minimal) geodesic of thesphere. Let us compute

Re〈∆(0),∆(1)〉 = Re〈ξ, π(eZe)ξ〉= 〈ξ, ξ〉 + 1

2〈ξ, Z2eξ〉 +14!〈ξ, Z4eξ〉 + . . .

= 〈ξ, ξ〉 + 12〈ξ, eZ2eξ〉 +

14!〈ξ, eZ4eξ〉 + . . .

= 1− 12‖Z‖2 +

14!‖Z‖4 − · · · = cos(‖Z‖);

observe that e and Z2 commute because Ze = (1− e)Z. Therefore ∆ is a minimizinggeodesic of SH .

On the other hand, note that ‖Ω(t)‖ = ‖π(Γte)ξ‖ 6 ‖Γte‖ = ‖γ(t) ⊗ x0‖ = ‖γ(t)‖.Therefore

length (γ) > length (Ω) > length (∆) = length (δ)

and our claim is proven. This proof is a mere adaptation of the argument in the mainresult of [27].

In the horizontal space Hx0 we have considered the vectors Z (Ze = (1 − e)Z)which are ‘tangent’ to the projective space Ee. The natural supplement in Hx0 forthe space of these vectors is the space V ∈LB(X)ah: eV e = V of normal vectors.Let Nx0

X = eV (x0) : V ∗ = −V, eV e = e, ‖V ‖ < π. As in the previous proposition, we

306 E. Andruchow, G. Corach and D. Stojanoffwill prove that Nx0

X is a complemented submanifold of SX and that certain geodesicsin Nx0

X are minimal. First note that Cx0X and Nx0

X are indeed orthogonal at x0.If c(t) ∈ Cx0

X and n(t) ∈ Nx0X are smooth curves with c(0) = n(0) = x0, then for t

small there exist smooth curves of vectors Zt, Vt in LB(X) with:

(i) Z∗t = −Zt and V ∗t = −Vt;(ii) Z0 = V0 = 0;(iii) Zte = (1− e)Zt and eVte = Vt;(iv) c(t) = eZt(x0) and n(t) = eVt(x0).

Then

〈c(0), n(0)〉 = 〈Z0(x0), V0(x0)〉 = 0.

Indeed, since x0〈Z0(x0), V0(x0)〉 = (x0 ⊗ Z0(x0))V0(x0) = [Z∗0 (x0 ⊗ x0)]∗V0(x0) and

[Z∗0 (x0 ⊗ x0)]∗V0 = [Z∗0 e]∗V0 = [(1− e)Z∗]∗V0 = Z0(1− e)V0 = 0

then x0〈Z0(x0), V0(x0)〉 = 0, and therefore 〈Z0(x0), V0(x0)〉 = 0. In other words, Cx0X

and Nx0X are 〈 , 〉-normal at x0.

If M is a metric space and m,m′ ∈M lie in the same connected component, then

dr(m,m′) = inf `(γ) : γ(t) ∈M with γ(0) = m and γ(1) = m′,where `(γ) is the length of γ and the infimum is taken over all rectifiablecurves γ joining m and m′. This metric is usually called the rectifiable metric. Putdr(m,m′) =∞ if m,m′ lie in different connected components.

If the space M is a differentiable manifold, one can compute dr using C1 curves.

Proposition 4·4. Nx0X is a C∞ submanifold of SX . Moreover

(a) Any two points x, y inNx0X such that the rectifiable distances dr(x, x0) and dr(y, x0)

are strictly less than π/2 can be joined by a unique minimal geodesic, which liesin Nx0

X .(b) If x ∈ Nx0

X satisfies dr(x, x0) < π, then there exists a unique geodesic curve joiningx and x0, which lies in Nx0

X and is shorter than any other curve in SX joining x0

and x.

Proof. First note that Nx0X is a submanifold. The space

eV e ∈LB(X):V ∗ = −V, V = eV e, ‖V ‖ < πis just the set of unitary elements eUe of the algebra eLB(X)e such that‖eUe− e‖ < 1 and is therefore a C∞ submanifold of this algebra. On the other handeLB(X)e is a complemented subspace of LB(X)e. Therefore, eV e ∈: V ∗ = −V, V =eV e, ‖V ‖ < π ⊂ LB(X)e is a submanifold. Since Nx0

X = −1x0

(eV e ∈ LB(X) :V ∗ = −V, V = eV e), our first claim is proven.

It has been already noted that if γ is a curve in SX , then the length of γ equalsthe length of any lifting Γ with the property that Γt(x0) = γ(t) (in fact it was shownin the case of the horizontal lifting, but the claim remains valid with any lifting).

Let now x, y ∈ Nx0X satisfy dr(x, x0) < π/2 and dr(y, x0) < π/2. Then x = eV (x0)

and y = eW (x0) with eV e = V, eWe = W and both ‖V ‖, ‖W‖ less than π/2. ThereforeeV e and eW e are unitaries in eLB(X)e lying at distance less than π and can be joinedby a minimal geodesic σ of this group. The curve σ is of the form σt = eV etZ for certain

Geometry of the sphere of a Hilbert module 307Z = eZe, Z∗ = −Z with ‖Z‖ < π and lies in U = eUe ∈ UeLB (X)e: ‖U − e‖ < 1(see [4]).

First we must see that σt(x0) is a geodesic of SX lying in Nx0X .

It clearly lies in Nx0X : since x0 (N

x0X ) = U = eUe ∈ UeLB (X)e: ‖U − e‖ < 1 and

σt = σte = σt(x0 ⊗ x0) = σt(x0) ⊗ x0 = x0 (σt(x0)). Let us see that it is a geodesic.Note that σt(x0) = eV (etZ(x0)) = eV (γ(t)), where γ(t) = etZ(x0) is clearly a geodesicof SX . The unitary group ULB (X) acts on SX carrying geodesics to geodesics and (a)is proven. Part (b) can be proven in an analogous way.

Remark 4·5. In the von Neumann algebra case (B, LB(X) von Neumannalgebras), then Proposition 4·4 holds taking eV e:V ∗ = −V, eV e = V = UeLB (X)e

instead of Nx0X , i.e. removing the condition ‖V ‖ < π in the definition of Nx0

X . Thiscondition is added in the C∗-algebra case in order that Nx0

X be a submanifold.

5. The index operator

In what follows,B is a von Neumann algebra and the Hilbert moduleX is assumedto be self-dual. Among the consequences of self-duality we point out that LB(X) is avon Neumann algebra and that the module X admits ‘orthonormal’ basis. Namely,one can find a set xii∈I in X such that 〈xi, xj〉 = 0 if i j; 〈xi, xi〉 = pi isa projection in B, and such that every element x ∈ X has a weakly convergentexpansion x =

∑i∈I xi〈xi, x〉 (see [18] or [23]).

In order to establish some topological properties of SX , it is useful to know whattype of algebra LB(X) is, knowing the type of B. The first result in this line wasobtained by Kaplansky in [15], where he showed that if B is abelian, then LB(X)is type I.

First recall that B and LB(X) have the same centre. More precisely, the centre ofLB(X) consists of operators which are right multiplication by elements in the centreof B. This allows one to consider spheres of modules over algebras B of a definitetype. Indeed, if p is a central projection in B, then X splits in the direct sum of themodules X;p over Bp and X;(1− p) over B(1− p). The sphere SX also splits in thecorresponding spheres

SX;p = x;p ∈ X;p :< x;p, x;p >= pand the analogous definition for SX;(1−p). Note also that LB;p(X;p) identifies withLB(X)p (here p denotes the adjointable operator ‘right multiplication by p’). There-fore, if we use the central projections of the type decomposition of B, we can restrictthe analysis to the case when B is of one particular type.

In [24, proposition 2·6, 2·7] Paschke proved that if X is a self-dual Hilbert modulewith orthonormal basis xii∈I then LB(X) is isomorphic to P (B⊗L(`2(I))P whereP ∈ B ⊗ L(`2(I)) is the diagonal infinite matrix with pi =< xi, xi > in the i, i-entryof the diagonal. This result implies that B is of type I, II or III if and only if LB(X)is of type I, II or III.

In particular, if B is finite, then LB(X) is semifinite.

Lemma 5·1. If B is infinite, then LB(X) is infinite.

Proof. Since B is infinite, there exists v ∈ B such that vv∗ = 1 and v∗v = p < 1.

308 E. Andruchow, G. Corach and D. StojanoffLet x ∈ SX . Then x⊗ x is an infinite projection in LB(X). Indeed, let V = x;v⊗ x,then

V ∗V = x〈x;v, x;v〉〈x, 〉 = x;p〈x, 〉 = x;p⊗ x;p < x⊗ xand

V V ∗ = x;v〈x, x;v〉〈x, 〉 = x⊗ x.Clearly x;p⊗x;px⊗x, since the former vanishes at X;(1−p) and the latter doesnot. q

The converse of this statement does not hold. Take for example B = C and Xa Hilbert space. However, if B is finite, then all projections of the form x ⊗ x forx ∈ SX are finite.

Proposition 5·2. If B is finite and x ∈ SX , then x⊗ x is finite in LB(X).

Proof. Let V ∈ LB(X) such that V ∗V = x ⊗ x and V V ∗ 6 x ⊗ x. Then V =V (x⊗x) = V (x)⊗x, and V V ∗ = (V (x)⊗x)(x⊗V (x)) = V (x)⊗V (x). The inequalityV V ∗ 6 x⊗ x implies that

V (x)⊗ V (x) = (x⊗ x)V V ∗ = x;〈x, V (x)〉 ⊗ V (x).

Since 〈V (x), V (x)〉 = 〈V ∗V (x), x〉 = 〈x, x〉 = 1, the above equality implies that V (x) =x;〈x, V (x)〉. Therefore

1 = 〈V (x), V (x)〉 = 〈V (x), x;〈x, V (x)〉〉 = 〈V (x), x〉〈x, V (x)〉.Since B is finite and 〈V (x), x〉∗ = 〈x, V (x)〉, we have also that

〈x, V (x)〉〈V (x), x〉 = 1.

Then

V V ∗(z) = (x;〈x, V (x)〉 ⊗ x;〈x, V (x)〉)(z) = x;〈x, V (x)〉〈x;〈x, V (x)〉, z〉= x;〈x, V (x)〉〈V (x), x〉〈x, z〉 = (x⊗ x)(z).

Suppose now that B is finite and let τ be the centre valued trace of B. There is anatural linear map defined from the space FB(X) of finite rank operators on LB(X),i.e. linear combinations of operators of the form x⊗ y for x, y ∈ X, onto the centreZ(B) of B. Namely

m : FB(X)→Z(B), m(x⊗ y) = τ (〈x, y〉).It is straightforward to verify that m is positive, tracial and faithful. Note that ifB = C then m is the usual trace for finite rank operators on the Hilbert space X.

Lemma 5·3. If B is finite and m is defined as above, then m extends to a faithful,normal and semifinite generalized trace on LB(X).

Proof. Let xii∈I be an orthonormal basis for X. If T ∈LB(X), put

φ(T ∗T ) =∑i∈I

τ (〈T (xi), T (xi)〉).

Then φ is a faithful and normal operator valued weight on LB(X), because the innerproduct 〈 , 〉 is weakly continuous. We claim that φ coincides with m on FB(X) and

Geometry of the sphere of a Hilbert module 309this ends the proof, because FB(X) is ultraweakly dense in LB(X) if X is self-dual(see [23]).

Let x ∈ X; then x =∑

i∈I xi〈xi, x〉. If T ∈LB(X), then T (x) =∑

i∈I T (xi)〈xi, x〉;in other words, T equals the (pointwise ultraweakly convergent) sum

∑i∈I T (xi)⊗xi.

By normality, it suffices to prove our claim for operators T which are finite sums∑nk=1 T (xjk )⊗ xjk . In this case

φ(T ∗T ) = τ

(∑i∈I

⟨n∑k=1

T (xjk )⊗ xjk (xi),n∑k=1

T (xjk )⊗ xjk (xi)

⟩)

= τ

(n∑k=1

〈T (xjk ), T (xjk )〉pjk)

= τ

(n∑k=1

〈T (xjk ), T (xjk )〉).

On the other hand

m(T ∗T ) = m

(n∑

l,k=1

xjk〈T (xjk ), T (xjl)〉 ⊗ xjl)

= τ

(n∑

l,k=1

〈T (xjl), T (xjk )〉〈xjk , xjl〉)

= τ

(n∑k=1

〈T (xjk ), T (xjk )〉),

which completes the proof. q

When is LB(X) is finite? Clearly the finiteness of B is not sufficient, as again thecase B = C shows.

Suppose that B is finite and let τ be the centre-valued trace of B. Let xii∈I bean orthonormal basis for X and pi =< xi, xi >, i ∈ I.

Denote by D the (possibly unbounded) positive operator affiliated to the centreof B,

D =∑i∈I

τ (pi). (5·4)

D does not depend on the choice of the basis of X; in fact, D = m(1), i.e. D is definedin terms of data independent of the particular choice of the basis xii∈I .

Proposition 5·5. IfB and LB(X) are finite, thenD is a densely defined, self-adjoint(positive), invertible operator and its inverse D−1 is a (bounded) element of Z(B).

Proof. By Lemma 5·3, m is a generalized trace of LB(X) and since 1 ∈LB(X) isa finite projection, by [30, V·2·35], D = m(1) defines a continuous positive functionwhich is finite on a open dense subset of the spectrum Ω of Z(B). Therefore D isa densely defined self-adjoint positive operator on L2(Ω, dµ) (for any probabilitymeasure µ in Ω). Denote by G the open dense subset of Ω where D ˆ is finite. Then ifχ ∈ G, since D ˆ(χ) is finite, χ m is a positive (bounded) functional in LB(X). Pickx ∈ SX , then 1 > x⊗ x, and therefore

χ m(1) > χ m(x⊗ x) = 1,

i.e. D ˆ(χ) > 1. Therefore 1/D ˆ is defined on the whole Ω (equalling zero at the pointswhere D ˆ is infinite), is continuous and positive. In other words, 0 6 D−1 6 1 inZ(B).

310 E. Andruchow, G. Corach and D. StojanoffNote that, with the hypothesis above, D−1 = Tr (x ⊗ x), where Tr is the centre-

valued trace of LB(X) and x is any element in SX .Also note that D > 1 and D = 1 implies that X is isomorphic to the trivial

B-module B. This operator D shall be subsequently called the index operator for X.In the case thatX is a von Neumann algebraA containingB and the inner product

is given by a finite index conditional expectation E : A→ B, then D = E([E]), where[E] is the (bounded) centre-valued index of E as defined in [6]. If [E], as in the factorcase, is a scalar, or more generally, if [E] lies in B (see [10] or [26]), then D coincideswith the index ofE. Under certain conditions, a tunnel of algebras can be constructedunder B. In this case D equals the index of the conditional expectation at the lowerlevel (see [10]).

Proposition 5·6. Under the current assumptions and notations, LB(X) is finite ifD is bounded.

Proof. Straightforward. q

The following example shows that the boundedness of D is not necessary forLB(X) to be finite. Let B = L∞(0, 1) and pn = χ[1−1/n,1] ∈ B for n ∈ N. LetXn = pnB and X = UDSXn: n ∈ N be the ultraweak direct sum of the modulesXn as defined in [23]. Then ([24]) LB(X) is isomorphic to P (B ⊗ L(`2))P with Pthe infinite diagonal matrix with pn in the n,n-entry. The operator D =

∑n∈N pn is

unbounded, because for each k ∈ N, Dpk > kpk. On the other hand LB(X) is finite,because the projection P is clearly finite in B ⊗ L(`2).

However, ifB is a factor and LB(X) is finite, then clearlyD is bounded, i.e. a finitepositive scalar. Indeed, in this case m must be a multiple of the trace of LB(X).

If B is finite and X is a finitely generated B-module, then LB(X) is finite. Theconverse of this may not hold. In [12] Jolissaint gave an example of a conditionalexpectation E : A→ B between finite von Neumann algebras such that E is of finiteindex but not of strongly finite index. As a consequence, A is a self-dual Hilbertmodule over B, which is not finitely generated. The algebra LB(A) is isomorphic toA1 ([6]), which is necessarily also finite ([12]).

However, the converse does hold if B is a (finite) factor.

Proposition 5·7. If B is a finite factor, then LB(X) is finite if and only if X isfinitely generated.

Proof. We only need to prove the only if part. We have that D =∑

i∈I τ (pi) ∈ R+

is finite. Note that this implies that I is countable. Since B is a factor, we can orderthe projections pin so that

pi1 pi2 pi3 . . .where denotes the Murray–von Neumann partial order for projections. Next, sincepik pik+1 there exists a projection qk+1 such that qk+1 6 pik and qk+1 is unitarilyequivalent to pjk+1 . Denote by uk+1 the unitary element of B implementing thisequivalence. If we replace the element xik+1 in the original orthonormal basis by theelement x′ik+1

= xik+1 .uk+1, then one obtains another orthonormal basis (and the sum∑τ (〈x′i, x′i〉) does not change). Therefore we may suppose the projections ordered

Geometry of the sphere of a Hilbert module 311by the inclusion of their ranges:

pi1 > pi2 > pi3 > . . .

If I were infinite, one could pick N arbitrarily large and choose a vector ξ of normone in the range of piN . Then

∑〈piξ, ξ〉 > N . On the other hand∑

i∈I pi = r is abounded element of B and therefore

∑i∈I〈piξ, ξ〉 6 ‖r‖, a contradiction.

6. The fundamental group of SX in the von Neumann algebra case

In this section, the results of [11], combined with the homogeneous structure ofSX as well as the index operator defined in the previous section, will be used tocompute the fundamental group of SX . In this task, one is readily confined to thecase when LB(X) is finite, because otherwise π1(SX) is trivial (see [11]).

First let us state this result concerning the connectedness of SX .

Proposition 6·1. Suppose that B is finite. If LB(X) is also finite then SX is con-nected. For example this is the case if D is bounded.

Proof. The proof proceeds as in Corollary 2·11.

Let us proceed with the computation of the fundamental group of SX . Fix x0 ∈ SX .The map

Πx0: ULB (X) → SX , Πx0 (U ) = U (x0)

is a principal bundle with structure group Ix0 = U ∈ ULB (X) : U (x0) = x0. Ife = x0 ⊗ x0, then Ix0 identifies with the unitary group of (1− e)LB(X)(1− e). Thisbundle gives rise to the exact sequence of the homotopy groups

· · · → π1(U(1−e)LB (X)(1−e), e)→ π1(ULB (X), 1)→ π1(SX , x0)→ 0.

If B is infinite, then so is LB(X), and therefore π1(ULB (X), 1) is trivial (see [11]).Then we have that if B is infinite or B is finite with LB(X) infinite, then π1(SX , x0)is trivial.

So it remains to consider the case whenB and LB(X) are finite. IfB (and thereforealso LB(X)) is of type II1, again by results from [11], π1(ULB (X), 1) is isomorphicto the additive group of continuous functions C(Ω,R) defined on Ω = Ω(Z(B))the spectrum of the common centre of B and LB(X). Handelmann (following [3])proved that π1(ULB (X), 1) is generated by the loops αP (t) = ei2πtP for P projectionsin LB(X). Moreover, the class of the loop αP is carried by the above isomorphism tothe continuous function Tr (P ), i.e. if χ ∈ Ω, Tr (P )(χ) = χ(Tr (P )), where Tr denotesthe centre-valued trace of LB(X).

In order to describe π1(SX , x0) we need to compute the image of the homomorphismλ : π1(U(1−e)LB (X)(1−e), 1− e)→ π1(ULB (X), 1) induced by the ‘inclusion’

U(1−e)LB (X)(1−e) → ULB (X), (1− e)w(1− e) 7→ e + (1− e)w(1− e).Clearly λ(π1(U(1−e)LB (X)(1−e), 1 − e)) is generated by the classes of loops αP withP 6 1−e. Therefore we need to identify the additive subgroup of C(Ω,R) generatedby the functions Tr (P ) : P 6 1− e.

Denote by Zx0 ⊂ Ω the set of zeros of 1− Tr (e).

312 E. Andruchow, G. Corach and D. StojanoffLemma 6·2. The image of the homomorphism λ is the subgroup of functions ofC(Ω,R)

which are multiples of 1− Tr (e).

Proof. Let f be a multiple of 1−Tr (e), f = g(1−Tr (e) ). Let us show that it belongsto the image of λ. Clearly, it suffices to consider g positive. Since g is bounded, thereexists n ∈ N such that (1/n)g 6 1. Then 0 6 (1/n)g(1−Tr (e) ) = (1/n)f 6 Tr (1−e).Therefore there exists a projectionQ in Z(B) such thatQ 6 1−e and Tr (Q) = (1/n)fand then f belongs to the image of λ.

In order to prove that the image of λ consists only of multiples of 1 − Tr (e),it suffices to prove that the functions h with 0 6 h 6 1 − Tr (e) are multiples of1−Tr (e). Let h be one such function. Then h/(1− Tr (e) ) is well defined, continuousand bounded (by 1) in the open set Ω − Zx0 . The closure (Ω − Zx0 )

− is a clopen inΩ, and is therefore a stonian space. By [30, III·1·8], h/(1− Tr (e) ) can be extendedto a continuous function g in this closure, and using Tietze’s theorem, it can befurthermore extended to a continuous function on the whole Ω. Therefore

g(1− Tr (e) ) = h,

because both functions coincide in Ω − Zx0 , and they both vanish in Zx0 .

This result in particular implies that if 1− Tr (e) is invertible then λ is surjectiveand therefore π1(SX , x0) is trivial. Such is the case, for example, if B is a II1 factor.In this case Ω consists of one point and 1 − Tr (e) (∈ R) is invertible provided thate 1 in LB(X). Note that e = 1 implies that X is isomorphic to B (as right Hilbertmodules).

If X = B then SX = b ∈ B : bb∗ = 1, and since B is finite this is just the unitarygroup of B, and then π1(SX , x0) = C(Ω,R).

In general, we have for the II1 case:

Proposition 6·3. If B and LB(X) are of type II1, then

π1(SX , x0) ' C(Ω,R)/f ∈ C(Ω,R) : f = g(1−D−1)'Z(B)s/z ∈Z(B)s: z = c(1−D−1), c ∈Z(B)s

where Z(B)s denotes the space of self-adjoint elements of Z(B), and the isomorphismsinvolved are (additive) group isomorphisms.

Proof. Note that the center valued trace Tr of LB(X) is the normalization of m(even if D is unbounded), i.e. Tr = D−1;m. Therefore Tr (e) = D−1, and the result isproven. q

Next, let us consider the type I case. Suppose B and LB(X) are finite and typeI. The next result follows as in the type II case and we omit the proof.

Proposition 6·4. If LB(X) is of type In, then

π1(SX , x0) ' C(

Ω,1nZ)/

f ∈ C(

Ω,1nZ)

: f = g(1−D−1) ).

Let us summarize the results concerning the fundamental group of the sphere SX .

Theorem 6·5. Let X be a seldual Hilbert module over a von Neumann algebra B

Geometry of the sphere of a Hilbert module 313and x0 ∈ SX . Denote by Pf the central projecion in LB(X) onto its finite part and letPf = ⊕n∈NPn ⊕ PII be the type decomposition of Pf . Then

π1(SX , x0) = C(ΩII,R)/〈(PII − PIID−1)ˆ〉 ×Πn∈NC

(Ωn,

1nZ)/〈(Pn − PnD−1)ˆ〉

where Ωn (resp. ΩII) is the Stone space of PnZ(B) (resp. PIIZ(B)) and 〈h〉 stands forthe subgroup of functions which are multiples of h by a continuous function.

Note that if B has atomic centre, then the part of π1(SX , x0) corresponding to thetype II central projection is trivial. Indeed, in this case SX.PII can be decomposedas a product of spheres over II1 factors, which have trivial fundamental groups.

Examples 6·6.(1) Recall the example after Proposition 5·6, B = L∞(0, 1), pn = χ[1−1/n,1], Xn =

Bpn and X = UDSXn : n ∈ N. Recall also that LB(X) = P (B ⊗ L(`2))P , forP the diagonal infinite matrix with pn in the n,n-entry. In order to compute thefundamental group of SX we must find the type decomposition of LB(X). Let enbe the characteristic function of the interval [(n− 1)/n, n/(n + 1)]. Straightforwardcomputations show that these projections give the type decomposition of LB(X).Namely, put Fn the diagonal infinite matrix with en in all its diagonal entries andEn = FnP . Then En ∈Z(LB(X)),

∑n∈NEn = 1 and

EnLB(X) ' L∞(n− 1n

,n

n + 1

)⊗Mn(C).

Moreover D−1 equals the constant 1/n on the interval [(n − 1)/n, n/(n + 1)], i.e.EnD

−1 = 1/nEn. Therefore

π1(SX , x0) ' Πn∈NC(Ω,Zn−1)

where Ω is the Stone space of L∞(0, 1).(2) Let M be a II1 factor and put B = M ⊗L∞(0, 1). Then Z(B) is isomorphic to

L∞(0, 1). For n ∈ N consider the function fn ∈ L∞(0, 1),

fn(t) =

0, if 0 6 t < n− 1n

11− t − n, if

n− 1n6 t < n

n + 1

1, ifn

n + 16 t 6 1.

Note that 0 6 fn(t) 6 1 and that∑

n∈N fn(t) = t/(1− t). Put now p0 = 1 and pn ∈ Bsuch that τ (pn) = fn, where τ is the centre-valued trace of B. Let P be the infinitediagonal matrix in B ⊗ L(`2) with p0, p1, p2, . . . in the diagonal. Then clearly P is afinite projection in B ⊗ L(`2). Therefore, if

X = UDSpnB : n ∈ N x 0then LB(X) = P (B ⊗ L(`2))P is of type II1 and

D =∞∑n=0

τ (pn) =1

1− t ,

314 E. Andruchow, G. Corach and D. Stojanoffi.e. 1 − D−1 = t. Then SX is connected and π1(SX) is isomorphic to the additivegroup

L∞R (0, 1)/tf (t) : f ∈ L∞R (0, 1).

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