Geodesics of projections in von Neumann algebras Esteban Andruchow∗ November 5, 2020 Abstract Let be a von Neumann algebra and A the manifold of projections in . There is a A P A natural linear connection in A, which in the finite dimensional case coincides with the the Levi-Civita connection of theP Grassmann manifold of Cn. In this paper we show that two projections p, q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ), if and only if A p q⊥ p⊥ q, ∧ ∼ ∧ where stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal∼ geodesic is unique if and only if p q⊥ = p⊥ q =0. If is a finite factor, any ∧ ∧ A pair of projections in the same connected component of A (i.e., with the same trace) can be joined by a minimal geodesic. P We explore certain relations with Jones’ index theory for subfactors. For instance, it is −1 shown that if are II1 factors with finite index [ : ] = t , then the geodesic N ⊂M M N 1/2 distance d(eN ,eM) between the induced projections eN and eM is d(eN ,eM) = arccos(t ). 2010 MSC: 58B20, 46L10, 53C22 Keywords: Projections, geodesics of projections, von Neumann algebras, index for subfac- tors. arXiv:2011.02013v1 [math.OA] 3 Nov 2020 1 Introduction ∗ If is a C -algebra, let A denote the set of (selfadjoint) projections in . A has a rich A P A P geometric structure, see for instante the papers [12] by H. Porta and L.Recht and [6] by G. ∞ Corach, H. Porta and L. Recht. In these works, it was shown that A is a C complemented P submanifold of , the set of selfadjoint elements of , and has a natural linear connection, As A whose geodesics can be explicitly computed. A metric is introduced, called in this context a Finsler metric: since the tangent spaces of A are closed and complemented linear subspaces of P , they can be endowed with the norm metric. With this Finsler metric, Porta and Recht [12] As showed that two projections p,q A which satisfy that p q < 1 can be joined by a unique ∈ P k − k geodesic, which is minimal for the metric (i.e., it is shorter than any other smooth curve in A P joining the same endpoints). ∗Instituto Argentino de Matematica,´ ‘Alberto P. Calderon’,´ CONICET, Saavedra 15 3er. piso, (1083) Buenos Aires, Argentina; and Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150 (1613), Los Polvorines, Argentina e-mail: [email protected] 1 In general, two projections p,q in satisfy that p q 1, so that what remains to consider A k − k≤ is what happens in the extremal case p q = 1: under what conditions does there exist a k − k geodesic, or a minimal geodesic, joining them. In the case when = ( ) the algebra of all bounded linear operators in a Hilbert space A B H , it is known (see for instance [4]) that there exists a geodesic joining p and q if and only if H dim R(p) N(q) = dim N(p) R(q). ∩ ∩ The geodesic is unique if and only if these intersections are trivial. The purpose of this note is to show that these facts remain valid if is a von Neumann A algebra, if we replace dim by the dimension relative to . Namely, it is shown that there exists A a minimal geodesic joining p and q in A, if and only if P p q⊥ p⊥ q. ∧ ∼ ∧ Here denotes the infimum of two projections, p⊥ = 1 p, and is the Murray-von Neumann ∧ − ∼ equivalence of projections. Also, it is shown that there exists a unique minimal geodesic if and only if p q⊥ =0= p⊥ q. ∧ ∧ We show that if is a finite factor, any pair of projections in in the same connected component A A of A (i.e., with the same trace), can be joined by a minimal geodesic. P In the final section of this paper, we explore the relationship with the index theory of von Neumann factors, introduced by V. Jones in [10]. A pairing of factors of type II , N ⊂ M 1 induces a sequence of projections, by means of the basic construction. We show that one recovers Jones index as a geodesic distance (minima of lengths of curves joining two given points): if e, f are two consecutive terms in the sequence of projections, then 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.