Geodesics of Projections in Von Neumann Algebras

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d(e, f) = arccos(t1/2), where t−1 = [ : ]. Also we show that if , with Jones’ projections e , e , M N N0 N1 ⊂ M 0 1 satisfy that e e < 1, then the unique geodesic δ(t) induces a smooth path of conditional k 0 − 1k expectations between and intermediate factors , and the parallel transport of this geodesic, M Nt induces a smooth path of normal -isomorphimsms between and . ∗ N0 Nt 2 Preliminaries

The space A is sometimes called the Grassmann manifold of . The reason for this name is P A that in the case when = ( ), B H parametrizes the set of closed subspaces of : to each A B H P ( ) H closed subspace corresponds the orthogonal projection PS onto . Let us describe below S⊂H S the main features of the geometry of A in the general case. P 2.1 Homogeneous structure ∗ ∗ Denote by A = u : u u = uu = 1 the unitary group of . It is a Banach-Lie group, U { ∈ A } ∗ A whose Banach-Lie algebra is as = x : x = x . This group acts on A by means of ∗ A { ∈ A − } P u p = upu , u A, p A. The action is smooth and locally transitive. It is known (see · ∈ U ∈ P [12], [6]) that A is what in diﬀerential geometry is called a homogeneous space of the group P

2 A. The local structure of A is described using this action. For instance, the tangent space U P (T A) of A at p is given by (T A) = x : x = px + xp . P p P P p { ∈As } The isotropy subgroup of the action at p, i.e., the elements of A which fix a given p, is U = v A : vp = pv . The isotropy algebra I at p is its Banach-Lie algebra I = y : Ip { ∈U } p p { ∈Aas yp = py . } It is useful, in order to describe and understand the geometry of A, to consider the diagonal P / co-diagonal decomposition of in terms of a fixed projection p A. Elements x which A 0 ∈ P ∈A commute with p , or equivalently, commute with the symmetry 2p 1, when written as 2 2 in 0 0 − × terms of p , have diagonal matrices. Co-diagonal matrices correspond with elements in which 0 A anti-commute with 2p 1. 0 − Then, the isotropy subgroup and the isotropy algebra , I at p , are respectively the sets Ip0 p0 0 of diagonal unitaries and diagonal anti-Hermitian elements of . On the other side, the tangent A space (T A) is the set of diagonal selfadjoint elements of . P p0 A 2.2 Reductive structure Given an homogeneous space, a reductive structure is a smooth distribution p H , 7→ p ⊂ Aas p A, of supplements of Ip in as, which is invariant under the action of p. That is, a ∈ P A I ∗ distribution H of closed linear subspaces of verifying that H I = ; vH v = H for p Aas p ⊕ p Aas p p all v ; and the map p H is smooth. ∈Ip 7→ p In the case of A, the choice of the (so called) horizontal subspaces H is natural. The P p 0 z ⊥ horizontal H defined in [6] is H = ∗ : z p p , i.e., the set of co-diagonal p p { z 0 ∈ A } anti-Hermitian elements of − A As in classical differential geometry, a reductive structure on a homogeneous space defines a linear connection: if X(t) is a smooth curve of vectors tangent to a smooth curve p(t) in A, P i.e., a smooth curve of selfadjoint elements of , which are pointwise co-diagonal with respect A to p(t), then the covariant derivative of the linear connection is given by

D X(t) := diagonal part w.r.t. p(t) of X˙ (t)= p(t)X˙ (t)p(t)+ p⊥(t)X˙ (t)p⊥(t). dt

It is not diﬃcult to deduce then that a geodesic starting at p A is given by the action of 0 ∈ P a one parameter group with horizontal (anti-Hermitian co-diagonal) velocity on p0 . Namely, 0 x given the base point p A, and a tangent vector x = ∗ (T A) 0 , the unique 0 ∈ P x 0 ∈ P p geodesic δ of A with δ(0) = p and δ˙(0) = x is given by P 0 tzx −tzx δ(t)= e p0e ,

0 x where zx := ∗ − . The horizontal element zx is characterized as the unique horizontal x 0 element at p0 such that [zx,p0]= x.

2.3 Finsler metric

As we mentioned above, one endows each tangent space (T A)p with the usual norm of . We P ∗A emphasize that this (constant) distribution of norms is not a Riemannian metric (the C -norm

3 is not given by an inner product), neither is it a Finsler metric in the classical sense (the map a a is non diﬀerentiable). Therefore the minimality result which we describe below does 7→ k k not follow from general considerations. It was proved in [12] using ad-hoc techniques.

1. Given p A and x (T A) , normalized so that x π/2, then the geodesic δ remains ∈ P ∈ P p k k≤ minimal for all t such that t 1. | |≤ 2. Given p,q A such that p q < 1, there exists a unique minimal geodesic δ such that ∈ P k − k δ(0) = p and δ(1) = q.

We shall call these geodesics (with initial speed x π/2) normalized geodesics. k k≤ 3 Von Neumann algebras

In this paper we consider the case when is a von Neumann algebra. We shall suppose A A acting in a Hilbert space (i.e., ( )). As we shall see, this representation is auxiliary, H A⊂B H and the results on the geometry of A do not depend on the representation. The main assertion P of this section is that the conditions of existence and uniqueness of minimal geodesics joining given projections p,q A are the a natural generalization of the conditions valid in the case ∈ P of ( ). B H ⊥ If p,q A, we denote by p = 1 p, and by p q the projection onto R(p) R(q) (which ∈ P − ∧ ∩ belongs to A); p and q are said to be Murray - von Neumann equivalent, in symbols p q, if P ∗ ∗ ∼ there exists v (a partial isometry) such that v v = p and vv = q. Our main result follows: ∈A Theorem 3.1. Let p,q A. ∈ P 1. There exists a geodesic δ of A joining p and q if and only if P p q⊥ p⊥ q. ∧ ∼ ∧ Moreover, the geodesic can be chosen minimal (i.e., normalized).

2. There is a unique normalized geodesic if and only if p q⊥ = p⊥ q = 0. ∧ ∧ Proof. Existence: suppose ﬁrst that p q⊥ p⊥ q. Consider following projections which sum ∧ ∼ ∧ 1 and commute both with p and q:

e = p q , e = p⊥ q⊥ , e = p q⊥ , e = p⊥ q , e = 1 e . 11 ∧ 00 ∧ 10 ∧ 11 ∧ 0 − i,j i,jX=0,1

It is straightforward to verify that eij commute with p and q, and thus e0 also does. The decomposition of the Hilbert space induced by these projections is sometimes called the Halmos decomposition of the space, in the presence of two closed subspaces (R(p) and R(q)); the last subspace R(e0), is called the generic part of p and q. We shall construct the exponent x of the geodesic joining p and q as a sum of anti-Hermitian elements in , A ′ ′′ x = x + x + x0,

′ ′′ where x acts in the range of e11 + e00, x acts in the range of e10 + e01 and x0 acts in the range of e0. Moreover, each of these elements is co-diagonal with respect to the corresponding

4 reduction of p to these subspaces. First note that peii = qeii (on e00 they are both zero, on e11 they are both the identity). Thus the exponent x′ can be chosen 0. Let us consider next the part in e . Here we make use of the representation ( ). 0 A⊂B H Denote by = R(e ), and by p = pe , qe the reductions of p,q to this subspace . Then, it H0 0 0 0 0 H0 is clear that p0,q0 lie in generic position ([9], [8]): their ranges and nullspaces intersect trivially. Thus, by a result by P. Halmos [9], there exist a Hilbert space , a positive operator X ( ) L ∈B L ( X /2 and a unitary isomorphism which carries k k ≤ P H0 →L×L 1 0 cos2(X) cos(X) sin(X) p to , and q to . 0 0 0 0 cos(X) sin(X) sin2(X)

Between these operator matrices, one can ﬁnd the (co-diagonal) exponent

0 X Z = , X 0 − which satisﬁes 1 0 cos2(X) cos(X) sin(X) eZ e−Z = . 0 0 cos(X) sin(X) sin2(X) These are straightforwward veriﬁcations, and provide the exponent for a geodesic joining the two operator matrices. One loses track though of how elements of are changed by the Halmos A isomorphism. The key fact to relate these matrices to the former projections p0,q0 is the following elementary identity proved in [4] 1 0 eZ = V, (1) 0 1 − where V is the unitary part in the polar decomposition of

1 0 cos2(X) cos(X) sin(X) B 1= + 1, − 0 0 cos(X) sin(X) sin2(X) − i.e. B 1= V B 1 . Again, this is an elementary matrix computation. Let b = p + q , and − | − | 0 0 0 let v0 be the isometric part in the polar decomposition (recall that e0 is the unit in this part of the algebra) b e = v b e . 0 − 0 0| 0 − 0| Clearly v and is carried by the Halmos isomorphism to V . Therefore, if one regards (1), 0 ∈ A it follows that the unitary element v (2p 1) is carried by this ismorphism to eZ , i.e. eZ 0 0 − corresponds to an element of . Moreover, A Z = X π/2, k k k k≤ which implies that Z is the unique anti-Hermitian logarithm of eZ with spectrum in ( iπ, iπ). − It follows that there exists a unique element x which corresponds to Z, and therefore 0 ∈ A satisﬁes x0 −x0 e p0e = q0. ′′ It remains to construct the exponent x acting in e10 + e01. Note that the reductions of p and q to this part are p(e + e )= p(p q⊥ + p⊥ q)= p q⊥, 10 01 ∧ ∧ ∧

5 ans similarly q(e + e )= p⊥ q. By hypothesis, there is a partial isometry w such that 10 01 ∧ ∈A w∗w = p q⊥ and ww∗ = p⊥ q. ∧ ∧ ′′ π ∗ ⊥ ⊥ ′′ Then x = i 2 (w + w ) does the feat: since p q p q, it follows that x is p10 + p01 ′′ ∧ ⊥ ∧ co-diagonal. Clearly x = π/2. Note also that w2 = 0, so that k k x′′ e = i(w + w∗).

Finally,

x′′ x′′ e (p q⊥)= i(w + w∗)(w∗w)= iww∗w = iww∗(w + w∗) = (p⊥ q)e , ∧ ∧ ′′ i.e., ex intertwines the reductions of p and q to this part. ′ ′′ If we put together x = x + x + x0, which is an orthogonal sum, we have a p-co-diagonal anti-Hermitian element of , with x π/2 (note that x′′ might be zero, if p q⊥ = p⊥ q = 0), A k k≤ ∧ ∧ which satisfies x x e pe− = q. Conversely, suppose that there exists a normalized geodesic wich joins p and q, i.e. there exists a p-co-diagonal anti-Hermitian element x with x π/2 such that expe−x = q. x ⊥ ⊥ ∈ A k kx ≤ We claim that e maps R(p q ) onto R(p q). Clearly e maps R(p) onto R(q). Pick ⊥ ∧ x ∧ ξ R(p q ) = R(p) N(q). Then e ξ R(q). It was noted in [12], that the fact that x is ∈ ∧ ∩ ∈ p-co-diagonal means that x anti-commutes with 2p 1. Thus, − x x (2p 1)e = e− (2p 1). − − Then, since (2p 1)ξ = ξ and (2q 1)ξ = ξ, − − − x x x x x x x (2p 1)e ξ = e− (2p 1)ξ = e− ξ = e− (2q 1)ξ = (2p 1)e− ξ = e (2p 1)ξ = e ξ, − − − − − − − − − i.e., exξ N(p), and thus ex(R(p) N(q)) R(q) N(p) The other inclusion follows similarly ∈ ∩ ⊂ ∩ (or by symmetry: in fact x is also q-co-diagonal, because x is the initial velocity of the reversed x ⊥− geodesic which starts at q). It folllows that w = e (p q ) is a partial isometry with initial ⊥ ⊥ ∧ ∈A space p q and final space p q. ∧ ⊥ ⊥ ∧ Uniqueness: if p q = p q = 0, then R(p) N(q)= N(p) R(q)= 0 , and there exists ∧ ∧ ∩ ∩ { } a unique normalized geodesic in B H joining p and q. By the first part of the proof, there is a P ( ) normalized geodesic joining them in A. Thus, it is unique. P Conversely, suppose that there exists a unique geodesic joining p and q. Then necessarily p q⊥ p⊥ q. Suppose that these projections are non zero. Then, there are infinitely many ∧ ∼ ∧ ∗ ⊥ ∗ ⊥ different partial isometries w such that w w = p q and ww = p q. As in the first part ∧ ′′ ∧ of the proof, any such w give rise to different exponents x , and thus different x, i.e. different geodesics joining p and q.

Remark 3.2. In the above result, it was shown in fact that the submanifold A B H is P ⊂ P ( ) totally geodesic: the geodesics of A are geodesics of the bigger manifold B H ; if p,q A are P P ( ) ∈ P joined by a unique geodesic of B H , then this geodesic remains inside A. P ( ) P

6 4 Hopf-Rinow theorem in ﬁnite factors

Two subspaces of dimension k in Cn can be joined by a minimal geodesic of the Levi-Civita connection in the Grassmann manifold. This fact can be proved using the projection formalism. That is, parametrizing subspaces with othogonal projections in Mn(C), by means of

n C PS M (C), ⊃ S ←→ ∈ n n where PS is the orthogonal projection onto . Two subspaces , C have the same dimen- S S T ⊂ sion if and only if the corresponding projections PS , PT have the same rank, i.e.

T r(PS PT ) = 0. − Let us see that in this case one has, automatically, that

dim( ⊥) = dim( ⊥ ). S∩T S ∩T This fact has an elementary proof. Let us prove it in a non totally elementary fashion, which will allow us to obtain a generalization. The operator A = PS PT is a selfadjoint contraction, − and if B = PS + PT ,

N(B 1) = ⊥ ⊥ = N(A 1) N(A + 1). − S∩T ⊕ S ∩T − ⊕ On the subspace N(B 1)⊥, B 1 is an invertible matrix, and the symmetry V in its polar − − ⊥ decomposition B 1 = V B 1 satisﬁes that V PS V = PT in N(B 1) . Then A is reduced − | − | − by N(B 1), and − V (A )V = A . N(B−1)⊥ − N(B−1)⊥

This implies that the spectrum of A is symmetric with respect to the origin: if λ is N(B−1)⊥ an eigenvalue of A with λ < 1, then λ is also an eignevalue of A, and they have the same | | − multiplicity: dim(N(A λ)) = dim(N(A + λ)). Then −

A = PN(A+1) + PN(A−1) + A = PN(A+1) + PN(A−1) + λ(PN(A−λ) PN(A+λ)). − N(B−1)⊥ − − 0<λ

Thus, the fact that T r(A) = 0, means that T r(PN(A−1))= T r(PN(A+1)). Thus PS and PT can be joined by a normalized geodesic. Remarkably, this geodesic is minimal for the Levi-Civita connection of the Grassmann manifold, but also, using the projection formalism, for the operator norm of Mn(C), the p-Schatten norms ([3]), or more generally, for unitary invariant norms (see [5]). Let us suppose now that is a ﬁnite von Neumann factor, with trace τ. We shall see that A the above argument holds (essentially unaltered):

Theorem 4.1. Let be a ﬁnite von Neumann factor with faithful normal trace τ. Two pro- A jections p,q A with p q (i.e., unitarily equivalent, or equivalently, in the same connected ∈ P ∼ component of A) can be joined by a normalized geodesic. P

7 Proof. Let a = p q, and again note that N(a 1) = R(p) N(q). Following previous notations, ⊥− −⊥ ∩ ′′ P − = p q = e . Similarly, P = p q = e . Therefore, P − = e + e := e . N(a 1) ∧ 10 N(a+1) ∧ 01 N(b 1) 10 01 Again, since e10 and e01 are eigenspaces of a, these projections reduce a, let a0 be the reduction of a to R(e′′)⊥. Then, we have that

a = e e + a . 10 − 01 0

The operator a0 is a diﬀerence of projections: a0 = p0 q0, where p0 and q0 are the reductions ′′ ⊥ − ′′ ⊥ of p and q to R(e ) , with N(a0 e0) = 0 (e0 is the identity in R(e ) ). It was shown by ± { } ∗ 2 Chandler Davis [7] that there exists a symmetry v0 (v0 = v0, v0 = e0; namely, v0 is the isometric part in the polar decomposition of b e ) such that 0 − 0 v a v = a . 0 0 0 − 0

Let µ be the projection-valued spectral measure of a0:

1 a0 = λdµ(λ). Z−1

As in the above argument in Mn(C), the existence of the symmetry v0 implies the symmetry of the spectral measure of a with respect to the origin: if Λ [ 1, 1] is a Borel subset, then 0 ⊂ − µ( Λ) = v µ(Λ)v . − 0 0 Then 1 τ(a0)= λdτ(µ(λ)) = 0, Z−1 because the function f(λ) = λ is odd and the measure τµ is symmetric with respect to the origin: τ(µ( Λ)) = τ(v µ(Λ)v )= τ(e µ(Λ)) = τ(µ(Λ)), − 0 0 0 because µ e . Then, since p q, ≤ 0 ∼ 0= τ(p) τ(q)= τ(a)= τ(e e + a )= τ(e ) τ(e ), − 10 − 01 0 10 − 01 so that τ(p q⊥)= τ(p⊥ q), i.e., p q⊥ p⊥ q. Therefore, by Theorem 3.1, there exists a ∧ ∧ ∧ ∼ ∧ (minimal) normalized geodesic joining p and q in A. P Remark 4.2. In [1], it was shown that in a ﬁnite algebra with faithful trace τ, the geodesics have minimal length also when measured with the ρ norms ρ of the trace, for ρ 2 ( x ρ = ∗ z −kz k ≥ k k (τ(x x)ρ/2)1/ρ). Namely, it was shown that if δ(t) = et pe t is a normalized geodesic ( z k k ≤ π/2) with δ(1) = q, and γ is any other smooth curve in A with γ(t )= p and γ(t )= q, then P 0 1 t1 ℓρ(γ) := γ˙ (t) ρdt ℓρ(δ)= z ρ. Zt0 k k ≥ k k

As we have seen, on ﬁnite factors, a version of the the Hopf-Rinow is valid in A, and the P geodesics are minimal for the usual norm of at every tangent space. However, as a consequence A of the fact in the above remark, we have that for the p-norms in the tangent space, including the

8 pseudo-Riemannian case p = 2, there are no normal neighbourhoods if is a type II factor. A 1 Indeed, for 2 ρ< , denote by ≤ ∞ d (p,q) = inf ℓ (γ) : γ is smooth and joins p and q in A ρ { ρ P } the metric induced in A by the ρ-norm. P Proposition 4.3. Let a type II factor and 2 ρ< Then there exist pairs of projections A 1 ≤ ∞ in A, which are arbitrarily close for the d metric, which can be joined by inﬁnitely many P ρ geodesics.

1 ⊥ Proof. Given 0 < r 2 , let p A such that τ(p) = r. Let q A such that q p ≤ ∈ P ⊥ ⊥ ∈ P ≤ and τ(q) = r (consider the reduced factor p p , which is also of type II1, and pick there a r A projection q with (renormalized) trace 1−r ). Then, the Halmos decomposition given by p and q yields (following the notation of the preceding section)

e00 = 0, e11 = 0, e10 = p, e10 = p, e01 = q, e0 = 0.

Since p q, there exist inﬁnitely many v such that v∗v = p and vv∗ = q. Any of these v ∼ ∈ A provides a geodesic joining p and q, given by (see the last part of the proof of Theorem 3.1) the π ∗ exponent x = i 2 (v + v ). The length of any of these geodesics is π π x = τ((x∗x)ρ/2)1/ρ = τ((v∗v + vv∗)ρ/2)1/ρ = 21/ρτ(p)1/ρ = π21/ρ−1r1/ρ. k kρ 2 2

5 Applications to ﬁnite index subfactors

V.F.R. Jones introduced the theory of index for subfactors of a II1 factor in [10]. An inclusion of II factors is said to be of finite index if the relative dimension (or coupling constant) N ⊂ M 1 2 −1 [ : ] := dimN (L ( ,τ)) = t M N M is finite (see [11]). A sequence of projections arises in this circumstance, by means of Jones’ basic cosntruction. Denote by τ the normalized trace of (and of the subsequent finite extensions M 2 2 which will be considered). Let eN be the orthogonal projection of L ( ,τ) onto L ( ,τ). M N This projection, restricted to L2( ,τ), induces the unique trace invariant conditional M ⊂ M expectation EN : . Jones proved that the von Neumann algebra 1 =< , eN > 2 M→N M M generated in (L ( ,τ)) by and eN is again a II factor, and that the inclusion B M M 1 M ⊂ M1 has finite index, with [ : ] = [ : ]. Thus, iterating the basic construction, a sequence M1 M M N of orthogonal projections arises: e1 = eN , e2 = eM,.... We shall be concerned only with the first two. These projections recover the index:

−1 τ(eN )= τ(eM)= t = [ : ] . M N In particular, they are unitarily equivalent in any factor of the tower of factors enabled by the basic construction, in which both lie. More precisely, Jones proved ([10], Proposition 3.4.1) that

⊥ ⊥ eN eMeN = teN and eN eM = eN eM = 0. ∧ ∧

9 It follows that eN and eM can be joined by a unique geodesic which lies in =< , eN , eM >. M2 M Thus, the ﬁnite index inclusion gives rise to a unique element zM N , the exponent N ⊂ M , ∈ M2 of this geodesic:

∗ zM N = zM N , d(eN , eM)= zM N π/2 , zM N is eN and eM co-diagonal, , − , k , k≤ , and zM N −zM N e , eN e , = eM. −1 The index [ : ]= t is related to the geodesic distance between eN and eM, meaured with M N the usual norm of , or with the ρ-norms (1 ρ< ): M2 ≤ ∞ Theorem 5.1. With the above notations,

1/2 1/ρ 1/2 d(eN , eM)= zM N = arccos(t ) and d (eN , eM)= zM N = t arccos(t ). k , k ρ k , kρ 2 ′ ′ Proof. The projections eN and eM act in L ( 1,τ). Denote by eN and eM the generic part of M′ 2 these projections, acting on the Hilbet space L ( 1,τ). By Halmos’ theorem, there exists ′ H ⊂ M ′ ′ an isometric isomorphism between and such that e and e are carried, respectively, H L×L N M onto 1 0 cos2(X) cos(X) sin(X) PN = and PM = , 0 0 cos(X) sin(X) sin2(X) where 0 X π/2. Note that since the only (non trivial) non generic part of eN and eM is ⊥ ⊥ ≤ ≤ ′ ′ ′ e e , on which both eN and eM act trivially, we have that e e e = eN eMeN = teN . N ∧ M N M N Therefore,

1 0 cos2(X) cos(X) sin(X) t = tPN = PN PMPN = , 0 0 cos(X) sin(X) sin2(X)

1/2 1/2 i.e., cos(X)= t 1L, and therefore X is a scalar multiple of the identity in : X = arccos(t )1L. L The unique exponent z = zM,N of the geodesic joining eN and eM, is zero on the non generic part e⊥ e⊥ , and in the generic part is related (via the Halmos’ isomorphism) to the operator N ∧ M 0 X Z = . X 0 − Then z∗z corresponds to

X2 0 1 0 Z∗ = = (arccos(t1/2))2 . 0 X2 0 1

∗ 1/2 2 Therefore z z = (arccos(t )) eN . The geodesic distance induced by the usual operator norm is ∗ 1/2 1/2 d(eN , eM)= z = z z = arccos(t ), k k k k and the one induced by the ρ norm is

1/2 1/ρ 1/ρ 1/2 d (eN , eM)= z = arccos(t )(τ(eN )) = t arccos(t ). ρ k kρ

10 Next, we consider the case of two projections arising from two subfactors , . N0 N1 ⊂ M These give rise to two orthogonal projections e , e in (L2( ,τ)). We make the assumption 0 1 B M that both inclusions have finite index: [ : ], [ : ] < . If both projections lie in M N0 M N1 ∞ the same II factor (with trace τ extending the trace of ), then a necessary and 1 M0 ⊃ M M sufficient condition for the existence of a geodesic joining e0 and e1 is τ(e0)= τ(e1). Lemma 5.2. Let , be finite index subfactors. Then there exists a II factor N0 N1 ⊂ M 1 M0 such that has finite index, and e , e . M ⊂ M0 0 1 ∈ M0 Proof. Let E : , i = 0, 1, be the unique trace preserving conditional expectations, i M→Ni giving rise to the orthogonal projections e , e . Let =< , e >, and F : the 0 1 M1 M 0 M1 → M corresponding expectation. Note that F = E F : is a conditional expectation, which 1 1 M1 →N1 is trace invariant (for the trace of ), and which corresponds to the finite index inclusion M1 : [ : ] = [ : ][ , ]. Let f be the orthogonal projection in (L2( )) N1 ⊂ M1 M1 N1 M1 M M N1 1 B M1 induced by this inclusion, and =< ,f >, which is a finite factor with M0 M1 1 [ : ] = [ : ][ , ] < . M0 M1 M1 M M N1 ∞ We claim that f = e . Denote by [x] the element x regarded as a vector in L2( ). 1 1 ∈ M1 M1 Then, if x , ∈ M f1([x]) = [F1(x)] = [E1(F (x))] = [E1(x)] = e1([x]). If ξ ⊥, then f (ξ)= e (ξ) = 0. ∈ M 1 1 ′′ 2 Remark 5.3. Note that 0 , e0, e1 (L ( 1)). However, , e0 and e1 act also M ⊂ {M ′′} ⊂ B M M on L2( ). Thus, the algebra , e , e (L2( )) is -isomomorphic (by means of a M {M 0 1} ⊂ B M1 ∗ normal isomorphism, given by restriction to L2( )) to the von Neumann II factor :=< M 1 M1,2 , e , e > (L2( )). M 0 1 ⊂B M Proposition 5.4. Let , be finite index subfactors. Then there exists a geodesic N0 N1 ⊂ M joining e and e if and only if [ : ] = [ : ]. 0 1 M N0 M N1 −1 Proof. [ : ] = [ : ]= t if and only if τM (e )= τM (e )= t. M N0 M N1 0 0 0 1 With the same notations as above, we have the following: Lemma 5.5. Suppose that e e < 1, and let δ(t) = etze e−tz, t [0, 1], be the unique k 0 − 1k 0 ∈ geodesic of M joining δ(0) = e and δ(1) = e . Then P 0 0 1 tz −tz E = δ M : := e e t t| M→Nt N0 ⊂ M is a pointwise smooth path of conditional expectations joining EN and EN (i.e., the map [0, 1] 0 1 ∋ t E (a) is C1 for all a ). 7→ t ∈ M ∈ M Proof. We shall use repeatedly the following argument. Suppose that m is normal, and ∈ M0 satisfies that m[ ] [ ], i.e., m as an operator acting in L2( ), leaves the dense linear M ⊂ M M manifold [ ] = [x] : x invariant, and let f be a continuous function in the spectrum M { ∈ M} σ(m) of m. Then f(m) also leaves [ ] invariant. Indeed, let pk(z, z¯) be polynomials in z andz ¯ M ∗ which converge uniformly to f(z) in σ(m). Clearly p (m,m ) leave [ ] invariant. Let x . k M ∈ M Then, if we denote by L the element x acting by left multiplication on L2( ), x M ∗ ∗ ∗ ∗ p (m,m )(x) p (m,m )(x) M = (p (m,m ) p (m,m ))L B 2 M k k − j k k k − j xk (L ( ))

11 ∗ ∗ ∗ ∗ p (m,m ) p (m,m ) B 2 M L B 2 M = p (m,m ) p (m,m ) B 2 M x . ≤ k k − j k (L ( ))k xk (L ( )) k k − j k (L ( ))k k It follows that p (m,m∗)(x) is a Cauchy sequence in , which converges to f(m)(x) . k M ∈ M Consider now the element e + e 1 . Note that e e < 1 implies that e + e 1 0 1 − ∈ M0 k 0 − 1k 0 1 − is invertible. Clearly, e + e 1 leaves [ ] invariant: 0 1 − M (e + e 1)([x]) = [E (x)+ E (x) 1] [ ] 0 1 − 0 1 − ∈ M for all x . By the above argument, it follows that e + e 1 −1 leaves [ ] invariant. Thus ∈ M | 0 1 − | M ez = (2e 1)(e + e 1) e + e 1 −1 0 − 0 1 − | 0 1 − | leaves [ ] invariant. On the other hand, as remarked before, the fact that e e < 1 also M k 0 − 1k implies that ez 1 < √2 < 2 (or equivalently, that z < π/2). It follows that there is k − k k k a continuous logarithm deﬁned in the spectrum of ez, arg : σ(ez) ( π/2, π/2). Therefore, → − again using the argument at the beginning of this proof, it follows that z leaves [ ] invariant. M Therefore, etz leave [ ] invariant for t [0, 1]. It follows that δ(t), restricted to , induce the M ∈ M linear mappings tz δ(t) M = e e e tz M : . | 0 − | M → M tz 2 −tz tz −tz The range of δ(t) M is e L ( )e = e e = . Clearly these maps are idem- | N0 ∩ M N0 Nt potents, -preserving, normal, and contractive for the norm of . Thus, by the theorem of ∗ M Tomiyama [13], they are normal conditional expectations, interpolating between E0 and E1. The fact that the path is strongly smooth is also clear.

Remark 5.6. Let us recall Theorem 2.6 of [2]: Let be a unital C∗-algebra and suppose that for t [0, 1] one has subalgebras 1 A ∈ ∈Bt ⊂A and conditional expectations E : . Assume that for each a , the map t E (a) t A→Bt ∈A 7→ t ∈A is continuously differentiable. Denote by dE : the derivative of E : dEt(a) = d E (a). t A→A t dt t For each fixed t, the operator dE : is bounded. Consider the differential equation, for t A→A a , ∈A α˙ (t) = [dE .E ](α(t)) t t . (2) α(0) = a

We call this equation the parallel transport equation. Denote by Γt the propagator of this equation, i.e., the map Γ : given by the solutions: Γ (a)= α(t) with α(0) = a. Then t A→A t • ΓtE0Γ−t = Et, and ∗ • Γ B : is a C -algebra isomorphism. t| 0 B0 →Bt Corollary 5.7. Let 0, 1 be subfactors with e0 e1 < 1. Then the exponentials tz N N ⊂ M tz k−tz − k Γt = e which induce the unique geodesic δ(t) = e e0e joining e0 and e1 in M0 ( 0 =< ′′ P M , e0, e1 > ), satisfy that M tz −tz Γ N : = e e t| 0 N0 →Nt N0 are normal -isomorphisms. In particular, and are isomorphic. ∗ N0 N1 tz Proof. It is straightforward to verify that e are the propagators Γt of equation (2) in this case: tz −tz since E = δ(t) M = e ME e M, we have that (the maps below are restricted to ): t | | 0 | M dE = zetzE e−tz etzE e−tzz t 0 − 0

12 and after straightforward computations

[dE , E ]= zetzE e−tz 2etzE zE e−tz + etzE e−tzz. t t 0 − 0 0 0 Since z is e co-diagonal, it maps L2( ) into L2( )⊥, and therefore E zE = 0. Thus, 0 N0 N0 0 0 tz tz tz tz [dEt, Et]e = ze E0 + e E0z = e (zE0 + E0z).

The fact that the element z is e0 co-diagonal, also means that

z = e z(1 e ) + (1 e )ze = e z 2e ze + ze = ze + e z. 0 − 0 − 0 0 0 − 0 0 0 0 0 Restricted to gives zE + E z = z. Therefore, for x , M 0 0 ∈ M tz tz tz tz · [dEt, Et]e x = e (zE0 + E0z)zx = e zx = (e x) .

tz tz That is, α(t) = e x is the solution of (2) with α(0) = x, i.e. Γ = e M is the propagator of t | this equation, and the proof follows using Theorem 2.6 of [2]

⊥ ⊥ Remark 5.8. Suppose that e0, e1 as above satisfy the condition e0 e1 =0= e0 e1 (weaker ∧ − ∧ that e e < 1), then there exists a unique geodesic δ(t) = etze e tz joining e and e . k 0 − 1k 0 0 1 We would like to know if also in this case, the propagators Γt of the parallel transport equation induce as in the above case, a curve of automorphisms. Following the same argument as above, it amounts to knowing if the projections δ(t) induce conditional expectations onto the intermediate algebras etz e−tz. N0 References

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(Esteban Andruchow) Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez 1150, (1613) Los Polvorines, Argentina and Instituto Argentino de Matem´atica, ‘Alberto P. Calder´on’, CONICET, Saavedra 15 3er. piso, (1083) Buenos Aires, Argentina.