The manifold of finite rank projections in the algebra L(H) of bounded linear operators

José M. Isidro and Michael Mackey

Facultad de Matemáticas, Universidad de Santiago, Santiago de Compostela, Spain.

                   E-mail: jmisidro@zmat.usc.es

Department of Mathematics, University College, Dublin 4, Ireland.

                   E-mail: Michael.Mackey@ucd.ie

Abstract. Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V = L(H). Using the algebraic structure of V, a torsionfree affine connection ∇ (that is invariant under the group of automorphisms of V) is defined on every connected component 𝔐 of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0 ≤ r ≤ ∞). We prove that 𝔐 admits a Riemann structure if and only if 𝔐 consists of projections that have the same finite rank r or the same finite corank, and in that case ∇ is the Levi-Civita and the Kähler connection of 𝔐. Moreover, 𝔐 turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed.

Keywords. JBW-algebras, Grassmann manifolds, Riemann manifolds.

AMS 2000 Subject Classification. 48G20, 72H51.

1. Preliminaries on JB-algebras

1.1. Introduction.

In [4] Hirzebruch proved that the manifold of minimal projections in a finite-dimensional simple formally real Jordan algebra is a compact Riemann symmetric space of rank 1, and that any such space arises in this way. Later on, in [14] Nomura established similar results for the manifold of minimal projections in a topologically simple JH-algebra (a real Jordan-Hilbert algebra). The results in [1], [5] and [6] lead to the idea that the structure of a JBW-algebra V might encode information about the differential geometry of some manifolds naturally associated to it [11]. In particular, that the knowledge of the JBWstructure of V is sufficient to study the manifold of projections in V. Every JBW-algebra can be decomposed into a sum of closed ideals V = V_{I} ⊕ V_{II} ⊕ V_{III} of types I, II, and III respectively, and for our purpose it is not a hard restriction to assume that V is irreducible. JBW-algebras of type III are not well understood. A typical example of

Supported by Ministerio de Educación y Cultura of Spain, Research Project PB 98-1371.

a type II JBW-algebra is L^{∞}[0, 1] whose lattice of projections is modular. Namely, it consists of characteristic functions of Lebesgue measurable subsets of [0, 1] and form a discrete topological space since we have ∥χ_{1} − χ_{2}∥ = 1 whenever χ_{1} ≠ χ_{2}. Thus we have to consider JBW-algebras of type I and we shall assume them to be factors, hence factors of type I_{n} for some cardinal number 1 ≤ n ≤ ∞. Those of type I_{2}, called spin factors, are Hilbert spaces ([2] th. 6.1.8) hence they are included in the work of Nomura. Factors of type I_{n} with 3 ≤ n < ∞ are certain spaces of matrices ([2] th. 5.3.8) and so they are included in the work of Hirzebruch. Thus essentially we have to consider the JBW-algebra V : = L(H)_{sa} of the selfadjoint operators on a Hilbert space H over some of fields ℝ, ℂ, ℍ ([2] th. 7.5.11). Here we make such a study in a systematic manner without the use of any global scalar product in V, in the complex case. With minor changes it applies to the other two fields.

The set M of all projections in V can be identified with the set of all closed subspaces of H, which is a Grassmann manifold 𝔾(H) in a classical way [9]. It is known that 𝔾(H) has several connected components 𝔐, each of which consists of projections p in V that have a fixed rank r, 0 ≤ r ≤ ∞. An affine connection ∇, that is invariant under the group Aut^{∘}(V) of automorphisms of V, is then defined on each connected component 𝔐 with only the help of the JBW-structure. With it, 𝔐 becomes a symmetric totally geodesic real analytic manifold. Moreover, it is possible (and in fact easy) to integrate the differential equation of the geodesics corresponding to initial conditions defined by purely algebraic equations. For r = 1, 𝔐 is the complex projective space ℙ(H).

Motivated by the above, we ask whether it is possible to define a Riemann structure on 𝔐, and a necessary and sufficient condition for this to happen is established. The tangent space to 𝔐 at a point p is the range of the \frac{1}{2}-Peirce projector of V at p, and 𝔐 admits a Riemann structure (if and) only if P_{1/2}(p)V is (homeomorphic to) a Hilbert space. In [6] it has been proved that the latter occurs if and only if the rank of p or the corank of p (the rank of 1 −p, where 1 is the unit of the algebra V) is finite. M is an ortho-complemented lattice, and the mapping p ↦ p^{⊥}: = 1 −p is an involutory homeomorphism. In fact, this involution is a real analytic diffeomorphism of M, hence it suffices to study the connected manifolds 𝔐 with r < ∞ which leads us again to the work of Nomura. Namely, consider the algebras of finite rank operators, of Hilbert-Schmidt operators, of compact operators, and of bounded operators on H, respectively, and the inclusions

F(H) ⊂ L_{2}(H) ⊂ L(H)_{0} ⊂ L(H)

The first three of these algebras have the same set of projections, which is exactly the set of finite rank projections in L(H). However, the topologies induced on F(H) by L_{2}(H) and L(H) do not coincide (unless dim H < ∞), and a priori there is no reason to expect that they should coincide on the set of projections (we shall see that this happens).

We then study the Riemann manifolds 𝔐 for r < ∞ without the help of any global scalar product in these algebras. A scalar product in the tangent bundle to 𝔐 is needed, of course, but it is locally provided in a canonical way by the JBW-algebra structure of V. We begin with a discussion of the subalgebras V [a, b] generated in V by certain pairs of elements (a, b). These subalgebras, that play a fundamental role in our study, turn out to be Jordan isomorphic to Sym (ℝ, 2), the algebra of 2 × 2 symmetric matrices with real entries and the usual Jordan matrix product, and therefore they are finite dimensional. By choosing an appropriate basis in V [a, b] it is easy to integrate the differential equations of the geodesics in 𝔐.

1.2. Preliminaries on JBW-algebras.

A Jordan algebra V is an algebra over ℝ or ℂ in which the following two identities hold for all x, y in V:

xy = yx, x^{2}(xy) = x(x^{2}y) (1)

Let V be a Jordan algebra. Then L(x) and P (x), (x ∈ V), are defined by

L(x)y: = xy,  P (x)y: = 2L(x)^{2}y − L(x^{2})y,  (y ∈ V). (2)

An element a ∈ V is an idempotent if a^{2} = a. If V has a unit 1 , then every idempotent a ∈ V gives rise to a vector space direct sum decomposition of V, the Peirce decomposition:

V = V_{1}(a) ⊕ V_{1/2}(a) ⊕ V_{0}(a), V_{k}(a): = {x ∈ V : ax = kx},

where k ∈ \{1, \frac{1}{2}, 0\} and the corresponding projectors E_{k}(a): V → V_{k}(a), called Peirce projectors, are given by

E_{1}(a) = P (a), E_{1/2}(a) = 2L(a) − 2P (a), E_{0}(a) = I − 2L(a) + P (a). (3)

If the idempotent a is fixed and no confusion is likely to occur, we write V_{k}: = V_{k}(a) and P_{k}: = E_{k}(a) for k ∈ \{1, \frac{1}{2}, 0\}. The Peirce multiplication rules hold

\begin{matrix}V_{0}V_{0} ⊂ V_{0},  V_{0}V_{1} & = \{0\},  V_{1}V_{1} ⊂ V_{1}, \\ (V_{0} ⊕ V_{1})V_{1/2} ⊂ V_{1/2}, & V_{1/2}V_{1/2} ⊂ V_{0} + V_{1}. \end{matrix} (4)

In particular, V_{0} and V_{1} are Jordan subalgebras of V, and [L(x) L(y)] = 0 for x ∈ V_{0} and y ∈ V_{1} with the usual commutator product [ , ].

A JB-algebra is a real Jordan algebra with a complete norm such that the following conditions hold

∥xy∥ ≤ ∥x∥ ∥y∥,  ∥x^{2}∥ = ∥x∥^{2},  ∥x^{2}∥ ≤ ∥x^{2} + y^{2}∥.

A JB^{∗}-algebra is a complex Jordan algebra U with an algebra involution^{∗}: x ↦ x^{∗} and a complete norm such that the following conditions hold

∥xy∥ ≤ ∥x∥ ∥y∥,  ∥x^{∗}∥ = ∥x∥,  ∥{xxx}∥ = ∥x∥^{3},

where the triple product {abc} is defined by {abc}: = (ab^{∗})c − (ca)b^{∗} + (b^{∗}c)a and satisfies the Jordan identity

{x{abc}y} = {{bax}cy} − {ba{xcy}} + {{bay}cx}. (5)

The operators x□y ∈ L(U) are defined by z ↦ x□y(z): = {xyz} for z ∈ U.

An element a ∈ U is selfadjoint if a^{∗} = a. The the set of them, denoted by U_{s}, is a JBalgebra. Conversely, if V is a JB-algebra then there is a unique Jordan algebra structure in U: = V ⊕ ı V, the complexification of V, such that U_{s} = V and there is a unique norm in U that converts it into a JB^{∗}-algebra [18]. We refer to U as the hermitification of V. The set Aut(U), of all Jordan algebra^{∗}-automorphisms of U, consists of surjective linear isometries of U and is a topological group in the topology of uniform convergence over the unit ball of U. By Aut^{∘}(U) we denote the connected component of the identity in Aut(U). Every element in Aut^{∘}(U) preserves the real subspace V and is uniquely determined by this restriction.

Let U be a JB^{∗}-algebra. We write Pro(U) for the set of self-adjoint idempotents in U and Tri(U): = {a ∈ U: {a, a, a} = a} for the set of tripotents in U. Clearly Pro(U) ⊂ Tri(U), and every non zero a ∈ Tri(U) satisfies ∥a∥ = 1. Two elements a, b ∈ U are orthogonal if ab = 0. A projection a ∈ U is said to be minimal if a ≠ 0 and P_{1}(a)U = ℂ a, and we let Min(U) denote the set of them. For a JB^{∗}-algebra it may occur that Min(U) = ∅.

A JBW-algebra is a JB-algebra whose underlying Banach space V is a dual space, which occurs if and only if the hermitification U = V ⊕ ıV is a dual Banach space. In that case U is called a JBW^{∗}-algebra, the predual U_{∗} of U is uniquely determined and σ(U, U_{∗}), the weak* topology on U, is well defined. Let U be the JBW^{∗}-algebra U: = L(H) of bounded linear operators z: H → H on a Hilbert space H. Then U is a unital algebra with plenty of projections each of which admits a representation of the form

a = Σ_{i∈I}a_{i}, convergence in the weak* topology,

for some indexed family of pairwise orthogonal minimal projections. The minimal cardinal of the set I is the rank of a and rank(a) = 1 if and only if a is minimal. The rank of the algebra U is the rank of its unit element.

A JB-algebra V is algebraically (resp. topologically) simple if {0} and V are its only ideals (resp. closed ideals).

Although not surveyed here, we shall occasionally use some relationships between JB^{∗}- algebras and their associated JB^{∗}-triples. Our main reference for JBW-algebras, JBW^{∗}- algebras and JB^{∗}-triples are [2] and [17].

1.3. Manifolds of projections in a JBW-algebra.

Let V be a JBW-algebra and denote by U: = V ⊕ ıV its hermitification. Then U is a JBW^{∗}-algebra and Pro(U) = Pro(V). In the Peirce decomposition of U induced by a ∈ Pro(U), the Peirce spaces are selfadjoint, that is U_{k}(a)^{∗} = U_{k}(a), and we have V_{k}(a) = U_{k}(a)_{s} where P_{k}(a)_{|_{V}} is the Peirce projector of a in the algebra V. For every u ∈ V_{1/2}(a), the operator

G(a, u): = 2(u□a − a□u) ∈ L(U) (6)

is an inner derivation of the JBW^{∗}-algebra U and the operator-valued mapping t ↦ exp tG(a, u), (t ∈ ℝ), is a one-parameter group of automorphisms of U each of which preserves V. The set Pro(V), endowed with its topology as a subset of V, is not connected, namely 0 and 1 are isolated points. We let M(p) denote the connected component of p in Pro(V). It is known that M(p) is a real analytic manifold whose tangent space at the point a is V_{1/2}(a), a local chart being given by

z ↦ [exp G(a, z)] a, z ∈ N,

for z in a suitable neighbourhood N of 0 in V_{1/2}(a). As a consequence, all points in M(p) are projections that have the same rank as p. As proved in ([11] th. 4.4), M(p) can also be viewed as a holomorphic manifold whose tangent space at the point a is U_{1/2}(a) a local chart being given in a neighbourhood M of 0 ∈ U_{1/2}(a) by

u ↦ [exp 2u□a] a, u ∈ M.

For a projection p, the operator S(p): = Id − 2P_{1/2}(p) ∈ L(V), called the Peirce reflection, is a symmetry, that is, a selfadjoint involution of V. Namely we have S(p)x = x for x ∈ V_{1}(p) + V_{0}(p) and S(p)x = −x for x ∈ V_{1/2}(p). Besides S(p) ∈ Aut(V) and S(p)p = p, hence S(p) preserves the connected component M(p) which in this way is a symmetric manifold.

We let 𝔇 be the Lie algebra of all smooth vector fields on M(p). A vector field X is now locally identifiable with a real analytic function X: N ⊂ V_{1/2}(a) → V_{1/2}(a). We always consider V_{1/2}(a) as submerged in V. For a function X: M(p) → V, we let X_{a} denote the value of X at the point a ∈ M(p). By X′_{a} we represent the Fréchet derivative of X at a, thus X′_{a} is a continuous linear operator V_{1/2}(a) → V. For two vector fields X, Y ∈ 𝔇 we define

(∇_{X} Y)_{a}: = P_{1/2}(a)(Y′_{a}(X_{a})), a ∈ M(p). (7)

It is known that (X, Y) ↦ ∇_{X} Y is a torsionfree Aut^{∘}(U)-invariant affine connection on the manifold M(p). For every a ∈ M(p) and every u ∈ V_{1/2}(a) the curve γ_{a,u}(t): = [exp tG(a, u)] a is a ∇-geodesic that is contained in the closed real Jordan subalgebra of V generated by (a, u). Proofs can be found in [1].

The following key result is known. Theorem 1.1. Let U be a JBW^{∗}-algebra. Then for p ∈ Pro(U) the following conditions are equivalent: (i) U_{1/2}(p) is a reflexive space. (ii) U_{1/2}(p) is linearly homeomorphic to a Hilbert space. (iii) rankU_{1/2}(p) < ∞. For U: = L(H) these conditions are equivalent to (iv) rank(p) < ∞ or rank(1 − p) < ∞. Proof.

The equivalence (i) ⟺ (ii) ⟺ (iii) is known [9]. The statement concerning U = L(H) has been established in [6] as follows: From the expression of the Peirce projectors, we have for all x ∈ U

P_{1/2}(p)x = 2(p□p − P (p))x =
(px + xp) − 2pxp = px(1 − p) + (1 − p)xp.

Hence p U_{1/2}(p) ⊂ U_{1/2}(p) and x ↦ px is a continuous projector U_{1/2}(p) → p U_{1/2}(p). Similarly x ↦ xp is a continuous projector U_{1/2}(p) → U_{1/2}(p) p and since pU_{1/2}(p) ∩ U_{1/2}p = 0 we have a topological direct sum decomposition U_{1/2}(p) = X_{1}⊕X_{2}. Therefore U_{1/2}(p) is reflexive if and only if so are the summands. But X_{1} : = p U_{1/2}(p) = {px(1 − p): x ∈ U} is reflexive if and only if rank(p) < ∞ or rank(1 − p) < ∞ as we wanted to see. □

Suppose p ∈ Pro(U) is such that rank(p) < ∞. Since this condition is Aut^{∘}(U)- invariant, all projections a ∈ M(p) satisfy it too, and the tangent space U_{1/2}(a)_{s} = V_{1/2}(a) to M(p) at any point a ∈ M(p) is a Hilbert space. In ([1], prop. 9.12) one can find an explicit expression for an Aut^{∘}(U)-invariant scalar product whose norm is equivalent to ∥ · ∥ in U_{1/2}(a). Since M(p) is connected, that scalar product, denoted by ⟨·, ·⟩_{a}, is determined up to a positive constant coefficient that can be normalized by requiring that minimal tripotents should have norm one. We shall not go into details as no explicit expression of it will be used here. Definition 1.2. We refer to this Hilbert space norm as the Levi norm in U_{1/2}(a) and denote it by | · |_{a} Definition 1.3. We define a Riemannian metric g on M(p) by

g(X, Y)_{a}: = ⟨X_{a}, Y_{a}⟩_{a}, X, Y ∈ 𝔇M(p), a ∈ M(p).

Proposition 1.4. The affine connection in (7) is the Levi-Civita (respectively, the Kähler) connection on the real analytic (the holomorphic) manifold M(p). Proof.

Indeed, ∇ is compatible with g, that is

X g(Y, Z) = g(∇_{X}Y, Z) + g(Y, ∇_{X}Z), X, Y, Z ∈ 𝔇M(p).

Moreover ∇ is torsionfree, hence by ([12] th.1.8.11) ∇ is the unique Riemann connection on M(p). Remark that when M(p) is looked as a holomorphic manifold ∇ is hermitian, that is, it satisfies g(i Y, i Z) = g(Y, Z), therefore ∇ is the only Levi-Civita connection on M(p). On the other hand ∇_{X} i Y = i ∇_{X} Y, hence ∇ is the only hermitian connection on M(p). Thus the Levi-Civita and the hermitian connection are the same in this case and so ∇ is the Kähler connection on M(p). Theorem 1.5. The Jordan-Banach algebra U_{0} : = L_{0}(H) of compact operators and U_{2} : = L_{2}(H), the Jordan-Hilbert algebra of Hilbert-Schmidt operators on H, have the same set of projections and induce on it the same topology. Proof.

The first assertion is clear, and Pro(U_{0}) = Pro(U_{2}) is precisely the set of finite rank projections in L(H). It is also clear the above algebras have the same set of projections of a given rank r, (1 ≤ r < ∞), say 𝔓. Denote by 𝔓_{0} and 𝔓_{2} the Banach manifold structures defined on 𝔓 according to our method and to Nomuras’ method, respectively. The corresponding tangent spaces at a are

T_{a}𝔓_{0} = {x ∈ U_{0} : 2ax = x}, T_{a}𝔓_{2} = {y ∈ U_{2} : 2ay = y}.

Thus we have T_{a}𝔓_{2} ⊂ T_{a}𝔓_{0}. On the other hand, a itself is a Hilbert-Schmidt operator and as U_{2} is an operator ideal, from x ∈ U_{0} and x = 2ax we get x ∈ U_{2}, hence T_{a}𝔓_{2} = T_{a}𝔓_{0} as vector spaces. But both T_{a}𝔓_{2} and T_{a}𝔓_{0} are JB^{∗}-triples (subtriples of U_{2} and U_{0}, respectively) with the same triple product. It is known that if a Banach space X admits a JB^{∗}-triple structure, then the triple product determines the topology of X in a unique way. This in our case implies that the topologies induced by U_{2} = L(H) and U_{0} = L_{0}(H) on the tangent space T_{a}𝔓 coincide. But then also coincide the topologies induced on 𝔓 by these two algebras as they are locally homeomorphic to the same Banach space. □

2. Equations of the geodesics.

For any Jordan algebra V, we let V [u, v] denote the subalgebra of V generated by (u, v). By 𝕊 := Sym (ℝ, 2) we denote the Jordan algebra of the symmetric 2 × 2 matrices with real entries and the usual Jordan matrix product. Recall that in 𝕊 the set Pro(𝕊) consists of the isolated points 0, 1 , and the one-parameter family of minimal projections

(8)  B(θ) := (\begin{matrix}cos^{2} θ & \frac{1}{2} sin 2θ \\ \frac{1}{2} sin 2θ & sin^{2} θ \end{matrix}) ,  θ ∈ ℝ.

The elements A := B(0) and C := B(\frac{π}{2}) satisfy A ∘ C = 0 and A + C = 1 , where 1 is the unit of the algebra 𝕊. The element

X := (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ∈ 𝕊_{1/2}(A) ∩ 𝕊_{1/2}(C) (9)

is a non zero tripotent and {A, X, C} is a basis in Sym (ℝ, 2).

Theorem 2.1. Let V be a unital Jordan algebra and let a ≠ 0 and u denote, respectively, a projection in V and a tripotent in V_{1/2}(a) such that au^{2} = a. Then for V [a, u] the following conditions hold: (i) V [a, u] = Span {a, u, u^{2}}. (ii) c := u^{2} − a is an idempotent such that ac = 0. (iii) u^{2} = a + c is the unit in V [a, u]. (iv) There is a unique Jordan isomorphism ψ_{a,u}: V [a, u] → Sym (ℝ, 2) that takes a, u and c to A, X and C respectively. Proof.

From u ∈ V_{1/2}(a) we get au = \frac{1}{2}u and by assumption au^{2} = a. Define c := u^{2} − a. Then the above results and the fact that u is a tripotent give

c^{2} = (u^{2} − a)^{2} = u^{2} − 2au^{2} + a^{2} = u^{2} − a = c.

Therefore c is an idempotent and ac = a(u^{2} − a) = au^{2} − a = 0. Moreover

cu^{2} = (u^{2} − a)u^{2} = u^{2} − u^{2}a = u^{2} − a = c

All this is collected in the following table

∘ a u u^{2} c
a a \frac{1}{2}u a 0
u · u^{2} u \frac{1}{2}u
u^{2} · · u^{2} c
c ⋯ c

which proves that the linear span of the set {a, u, u^{2}, c} is closed under the operation of taking Jordan products, and that u^{2} = a + c is the unit of V [a, u] Now it is clear that there is a unique Jordan isomorphism ψ_{a,u}: V [a, u] → Sym (ℝ, 2) with the desired conditions. □ Remark 2.2. For a tripotent u ∈ V_{1/2}(a), the minimality of a is a sufficient (but not necessary) condition for {auu} = au^{2} = a to be true.

Indeed, by the Peirce rules u^{2} ∈ V_{0}(a) ⊕ V_{1}(a) and V_{0}(a) V_{1}(a) = {0}, hence au^{2} ∈ V_{1}(a) = ℝa by the minimality of a, therefore au^{2} = ρa. Multiplication by u, the fundamental identities (1) and u^{3} = u yield

u(au^{2}) = ρua = \frac{ρ}{2}u u(au^{2}) = (ua)u^{2} = \frac{1}{2}u^{3} = \frac{1}{2}u,

hence \frac{1}{2}u(ρ − 1) = 0 and so ρ = 1 since u ≠ 0. Thus \{auu\} = au^{2} = a. □

For pairs (a, u) consisting of a projection a ≠ 0 and a tripotent u ∈ V_{1/2}(a) with au^{2} = a it is quite easy to obtain the equation of the geodesic γ_{a,u}(t) as shown now. Let us define a new product in V via x ▪ y := {xay}. Then (V, ▪) is a unital Jordan algebra with unit a. For n ∈ ℕ and x ∈ V we let x^{(n)} denote the n-th power of x in (V, ▪). Note that x^{n} ≠ x^{(n)}. Theorem 2.3. Let V := L(H), and let a ≠ 0 and u respectively be a projection in V and a tripotent in V_{1/2}(a) such that au^{2} = a. If γ_{a,u} is the geodesic with γ(0) = a and γ̇(0) = u, then γ_{a,u}(ℝ) ⊂ V [a, u]. More precisely we have

(10)  γ_{a,u}(t) = (cos^{2} t) a + (\frac{1}{2} sin 2t) u + (sin^{2} t) u^{(2)},  t ∈ ℝ.

Proof.

Let G(a, u) := 2(u□a − a□u) ∈ L(V). We have γ_{a,u}(t) = [ exp tG(a, u) ] a for all t ∈ ℝ, hence γ_{a,u}(ℝ) is contained in the closed real linear span of the sequence (G(a, u)^{n} a)_{n∈ℕ}. We shall prove that the assumptions u^{3} = u and au^{2} = a yield G(a, u)^{n} a ∈ V [a, u] for all n ∈ ℕ.

We have

G(a, u) a = u ∈ V [a, u].

By assumption {auu} = au^{2} = a, hence

G(a, u) u = 2{uau} − 2{auu} = 2(u^{(2)} − a) ∈ V [a, u]
G(a, u) u^{(2)} = 2{u a u^{(2)}} − 2{a u u^{(2)}}.

By the Peirce multiplication rules u^{(3)} = {uau^{(2)}} ∈ {V_{1/2}(a) V_{1}(a) V_{0}(a)} = 0. The Jordan identity (5) and (2.2) give

{auu^{(2)}} = {au{uau}} = {u{auu}a} − {{uaa}uu} + {ua{uua}} =

\{uaa\} − \frac{1}{2}\{uuu\} + \{uaa\} = \frac{1}{2}u ∈ V [a, u]

and G(a, u) u^{(2)} ∈ V [a, u].

Note that a, u, u^{(2)} belong to different Peirce a-spaces, in particular they are linearly independent unless u or u^{(2)} vanish. We have assumed u ≠ 0 and if ψ is the isomorphism in (2.1), then ψ^{−1}u^{(2)} = ψ^{−1}{uau} = {XAX} = C ≠ 0. Therefore they form a basis of V [a, u] and G(a, u) V [a, u] ⊂ V [a, u]. As a consequence γ_{a,u}(t) has a unique expression of the form

γ_{a,u}(t) = f_{1}(t)a + f_{1/2}(t)u + f_{0}(t)u^{(2)}, t ∈ ℝ,

for suitable real analytic scalar valued functions f_{k}(t), (k = 0, 1/2, 1). By taking the derivative with respect to t and replacing the expressions previously obtained for G(a, u) z with z ∈ {a, u, u^{(2)}}, we get

f′_{1}(t)a + f′_{1/2}(t)u + f′_{0}(t)u^{(2)} = γ̇_{a,u}(t) = G(a, u)(γ_{a,u}(t)) =
f_{1}(t)G(a, u)a + f_{1/2}(t)G(a, u)u + f_{0}(t)G(a, u)u^{(2)} =
−2f_{1/2}(t)a + (f_{1}(t) − f_{0}(t))u + 2f_{1/2}(t)u^{(2)},

whence we have the first order ordinary differential equation F′(t) = A(u)F (t) with the initial condition F (0) = (1, 0, 0), where A(u) is a 3 × 3 constant (that is, not depending on t) matrix and F (t) is the transpose of (f_{1}(t), f_{1/2}(t), f_{0}(t)). In fact

A := (\begin{matrix}0 & −2 & 0 \\ 1 & 0 & −1 \\ 0 & 2 & 0 \end{matrix}) .

One can easily check that the solution is the curve in (10). □

Motivated by this result, we shall now try to weaken the restrictions on u. Proposition 2.4. Let V, a and u respectively be a unital JB-algebra, a projection in V and a tripotent in V_{1/2}(a). Then the following conditions hold: (i) p := au^{2} is a projection that satisfies p ≤ a, u ∈ V_{1/2}(p) and pu^{2} = p. (ii) For u ≠ 0 we have p ≠ 0. (iii) If u and v are orthogonal as tripotents in V_{1/2}(a), then p := au^{2} and q := av^{2} are orthogonal as projections in V and pv = uq = 0. Proof.

To see that p is an idempotent it suffices to consider the subalgebra of V generated by a, u and the unit e, which is a special algebra. By the Peirce rules, u^{2} ∈ V_{1} ⊕V_{0} and so we may write u^{2} = x + y for x ∈ V_{1} and y ∈ V_{0}. Thus p = ax + ay = x ∈ V_{1} and pa = p. On the other hand, since u is a tripotent u^{2} = u^{4} and so using the Peirce rules again

p = au^{4} = ax^{2} + 2a(xy) + ay^{2} = x^{2} = p^{2}.

That is, p is a projection. Notice that (a−p)^{2} = a−p and so a−p ≥ 0. From u ∈ V_{1/2}(a) and the fundamental formulas (1) we get up = u(au^{2}) = (au)u^{2} = \frac{1}{2}uu^{2} = \frac{1}{2}u hence u ∈ V_{1/2}(p) and p ≠ 0 if u ≠ 0. To complete (i), we notice that

pu^{2} = p(x + y) = px = p^{2} = p

since p = x ∈ V_{1}(a) and y ∈ V_{0}(a).

Assume now that u□v = 0 and set p := au^{2}, q := av^{2}. (We recall that two tripotents u and v are orthogonal if and only if {u, v, v} = 0 or equivalently {u, u, v} = 0.) Then 0 = {uve} = uv and also u^{2}v = {u, u, v} = 0. Although u^{2} is a tripotent and vu^{2} = 0, we remark that two tripotents e and f which are orthogonal in the algebra sense (ef = 0) may not be orthogonal tripotents. However in our case

{u^{2}, u^{2}, v} = u^{4}v − (vu^{2})u^{2} + (vu^{2})u^{2} = uv = 0

and so u^{2} and v are orthogonal tripotents. In particular, u^{2}v^{2} = {u^{2}, v^{2}, 1} = 0.

Let

u^{2} = x + y ∈ V_{1}(a) + V_{0}(a), v^{2} = x′ + y′ ∈ V_{1}(a) + V_{0}(a)

as before. Then 0 = u^{2}v^{2} = xx′ + yy′ entails xx′ = yy′ = 0 and so

pq = (au^{2})(av^{2}) = [a(x + y)] [(x′ + y′)a] = (ax)(x′a) = xx′ = 0.

To complete the proof, consider the product pv = vp = v(u^{2}a) = L(v)L(u^{2})a. Since v operator commutes with u^{2} and v ∈ V_{1/2}(a), we have pv = L(v)L(u^{2})a = L(u^{2})L(v)a = u^{2}(va) = \frac{1}{2}u^{2}v = 0 as seen before. Similarly uq = 0. □

The following can be considered as a generalization of (2.1). Theorem 2.5. Let V := L(H), and let a and u respectively be a finite rank projection in V and a vector in V_{1/2}(a). Then we have a finite direct sum decomposition

V [a, u] ≈ V_{0} ⊕ ( ⨁_{1}^{s} V [a_{k}, u_{k}]) (11)

where a_{0} and a_{k}, (1 ≤ k ≤ s), are projections with a = a_{0} + Σa_{k}, u_{k} are tripotents in V_{1/2}(a_{k}), the subalgebras V [a_{k}, u_{k}] are pairwise orthogonal and a_{0}u = 0. Proof.

The hermitification U := V ⊕ iV of V is JBW^{∗}-triple and, by the Peirce rules, U_{1/2}(a) is a JBW^{∗}-subtriple which has finite rank by (1.1). Hence by [9] every element in U_{1/2}(a) has a unique spectral resolution, that is, a representation of the form z = ρ_{1}u_{1} +⋯+ρ_{s}u_{s} where 0 < ρ_{1} < ⋯ < ρ_{s}, the u_{k} are pairwise orthogonal (possibly not minimal) non zero tripotents in U_{1/2}(a) and s ≤ r = rankU_{1/2}(a) < ∞. If z is selfadjoint (that is, z ∈ V_{1/2}(a)), then the u_{k} in the spectral resolution of z are also selfadjoint. Indeed, u_{j}□u_{k} = 0 for j ≠ k, hence the successive odd powers z^{2l+1} of z are

z^{2l+1} = ρ_{1}^{2l+1}u_{1} + ⋯ + ρ_{s}u_{s}^{2l+1}, (0 ≤ l ≤ s − 1).

and the Vandermonde determinant det(ρ_{k}^{2l+1}) does not vanish since the ρ_{k} are pairwise distinct. Therefore the u_{k} are linear combinations with real coefficients of the powers z_{k}^{2l+1} ∈ V_{1/2}(a) and so u_{k} ∈ V_{1/2}(a).

Now we discuss the algebra V [a, u]. Let u = ξ_{1}u_{1} + ⋯ + ξ_{s}u_{s} with 0 < ξ_{1} < ⋯ < ξ_{s} be a spectral resolution of u, and let a_{k} := au_{k}^{2} for 1 ≤ k ≤ s. By (2.4) the projections a_{k} are pairwise orthogonal and satisfy

Σa_{k} ≤ a, u_{k} ∈ V_{1/2}(a_{k}), a_{k}u_{k}^{2} = a_{k} (1 ≤ k ≤ s).

Set

a_{0} := a − Σa_{k}, V_{0} := ℝ a_{0}. (12)

Hence dim V_{0} = 1 at most. We shall see that V [a, u] ≈ V_{0}⊕⨁ V [a_{k}, u_{k}] as an orthogonal direct sum. For that purpose consider the successive powers u^{l} of u which are given by u^{l} = ξ_{1}^{l}u_{1}^{l}+⋯+ξ_{s}^{l}u_{s}^{l}, (1 ≤ l ≤ s), since the u_{k} are pairwise orthogonal. A Vandermonde argument shows that u_{k} ∈ V [a, u] for 1 ≤ k ≤ s. Then a_{k} = au_{k}^{2} ∈ V [a, u] and a_{0} ∈ V [a, u]. Therefore V_{0}⊕⨁ V [a_{k}, u_{k}] ⊂ V [a, u] On the other hand, from a = a_{0}+Σa_{k} and u = Σξ_{k}u_{k} it follows V [a, u] ⊂ V_{0} ⊕ ⨁ V [a_{k}, u_{k}] whence we get (11) as soon as we show that the summands satisfy the required orthogonality properties .

We have already shown that a_{k}a_{j} = 0 = u_{k}u_{j} for all k ≠ j. By (2.4) we have a_{k}u_{j} = 0 = a_{j}u_{k} and so the subalgebras V [a_{k}, u_{k}] and V [a_{j}, u_{j}] are orthogonal for k ≠ j. It remains to prove that a_{0}u = 0. By assumption u ∈ V_{1/2}(a) and from u_{k} ∈ V_{1/2}(a_{k}) we get u ∈ V_{1/2}(Σa_{k}), hence

\frac{1}{2}u = au = (a_{0} + Σa_{k}) u = a_{0}u + (Σa_{k})u = a_{0}u + \frac{1}{2}u

which completes the proof. □ Corollary 2.6. Let V = L(H), and let a and u be respectively a finite rank projection in V and a vector in V_{1/2}(a). If a = a_{0} + Σa_{k} and u = Σu_{k} are the decompositions given in (2.5) then

[exp tG(a, u)] a = a_{0} + Σ_{k}[exp tG(a_{k}, u_{k})] a_{k}

Proof.

The linearity of G and orthogonality properties of the elements involved give

G(a, u) = G(a_{0}, u) + Σ_{k}G(a_{k}, u_{k}) = Σ_{k}G(a_{k}, u_{k}),
G(a_{k}, u_{k}) V [a_{j}, u_{j}] = 0, (k ≠ j).

Therefore G(a, u)^{n} = Σ_{k}G(a_{k}, u_{k})^{n} for all n ∈ ℕ, and the claim follows from the definition of exponential mapping. □

3. Geodesics connecting two given points.

Proposition 3.1. Let V be a JB-algebra and let a, b be two projections in V with {aba} = λa and {bab} = μb for some real numbers λ, μ. Then 0 ≤ λ = μ ≤ 1. Furthermore λ = 0 if and only if ab = 0, and λ = 1 if and only if a = b. Proof.

Projections are positive elements in V, hence {aba} ≥ 0 by ([2] prop. 3.3.6). Then {aba} = λa entails λ ≥ 0. But λ ≤ 1 since

λ = ∥λa∥ = ∥{aba}∥ ≤ ∥a∥^{2} ∥b∥ ≤ 1.

By ([2] lemma 3.5.2) we have ∥{ab^{2}a}∥ = ∥{ba^{2}b}∥ for arbitrary elements a, b in V, hence in our case

λ = ∥{aba}∥ = ∥{bab}∥ = μ.

Clearly λ = 0 is equivalent to aba = 0 which by ([2] lemma 4.2.2) is equivalent to ab = 0. For arbitrary projections p, q, the condition pqp = p is equivalent to p ≤ q, therefore λ = 1 yields aba = a and bab = b that is a ≤ b and b ≤ a and so a = b and conversely. □ Proposition 3.2. Let V be a unital Jordan algebra and let a, b ∈ V be two projections such that P (a)b = λa and P (b)a = λb hold for some real number 0 < λ < 1. Then for V [a, b] the following conditions hold: (i) V [a, b] = Span {a, b, ab}. (ii) There is a unique Jordan isomorphism ϕ_{a,b}: V [a, b] → Sym (ℝ, 2) that takes a, b and ab respectively to A, B(θ) and A ∘ B(θ). Proof.

Set p: = ab. It follows from Macdonalds’ theorem that

p^{2} = \frac{1}{2}a \{bab\} + \frac{1}{4}\{ab^{2}a\} + \frac{1}{4}\{ba^{2}b\}

hence in our case p^{2} = \frac{λ}{4}(a + b + 2p). The above results are shown in the following table

∘ a b  p
a a p \frac{1}{2}(p + λa)
b · b \frac{1}{2}(p + λb)
p · · \frac{λ}{4}(a + b + 2p)

This shows that the real linear span of the set {a, b, p} is closed under the operation of taking Jordan products, and so V [a, b] = ℝa ⊕ ℝb ⊕ ℝp. It is not difficult to check that {a, b, p} is a basis for V [a, b] hence dim V [a, b] = 3. The other assertion is now clear. □ Remark 3.3. The angle θ appearing in (3.2) can be expressed in terms of a, b. Indeed, since ϕ_{a,b} preserves triple products and {A, B(θ), A} = (cos^{2} θ)A we have λ = cos^{2} θ or cos^{2} θ = ∥P_{1}(a)b∥. Remark 3.4. The conditions P (a)b = λa and P (b)a = λb with 0 < λ < 1 are automatically satisfied by any pair of minimal projections a, b with a ≠ b and ab ≠ 0. However, minimality is not necessary in order to have them.

We use the isomorphisms ψ_{a,u}: V [a, u] → Sym (ℝ, 2) and ϕ_{a,b}: V [a, b] → Sym (ℝ, 2) to show that two distinct minimal projections a, b in V = L(H) can be joined by a geodesic in 𝔐. Theorem 3.5. Let V = L(H) and let 𝔐(1) be the set of minimal projections in V. If a, b in 𝔐(1) are such that a ≠ b, ab ≠ 0, and W := V [a, b] then there exists a tripotent u ∈ W_{1/2}(a) (unique up to sign) such that the geodesic t ↦ γ_{a,u}(t) connects a with b in 𝔐(1). Moreover, u is uniquely determined by the additional property b = γ_{a,u}(t) for some t > 0. Proof.

The pair of projections a, b determines uniquely the algebra V [a, b] and the Jordan isomorphism ϕ_{a,b}: V [a, b] → Sym (ℝ, 2) with the conditions in (3.2). In particular

ϕ(b) = (\begin{matrix}cos^{2} θ & \frac{1}{2} sin 2θ \\ \frac{1}{2} sin 2θ & sin^{2} θ \end{matrix})

for some θ with 0 < θ < \frac{π}{2}. Let u = ϕ_{a,b}^{−1}(X) where X, given in (9), is the unique (up to sign) tripotent in 𝕊_{1/2}(A). Note that au^{2} = a. By (2.3), γ_{a,u}(ℝ) ⊂ V [a, u] ⊂ V [a, b] is a geodesic whose image by the isomorphism ψ_{a,u}: V [a, b] → Sym (ℝ, 2) is

ψ(γ_{a,u}(t)) = (\begin{matrix}cos^{2} t & \frac{1}{2} sin 2t \\ \frac{1}{2} sin 2t & sin^{2} t \end{matrix})

Since G(a, u) is real linear on u, the definition of the exponential function gives γ_{a,ρu}(t) = γ_{a,u}(tρ) for all ρ and t ∈ ℝ. In particular γ_{a,−u}(t) = γ_{a,u}(−t). A glance to the above matrices shows that either b = γ_{a,u}(θ) or b = γ_{a, −u}(θ) where θ > 0. □ Corollary 3.6. With the notation and conditions in the statement of (3.5), there is a unique vector v in W_{1/2}(a) such that b = γ_{a,v}(1) where v = θ u for some tripotent u in W_{1/2}(a) with au^{2} = a and some θ with 0 < θ < \frac{π}{2}. Proof.

As proved before, we have b = γ_{a,u}(θ) for a uniquely determined tripotent u ∈ W_{1/2}(a) with au^{2} = a and the unique real number θ given by cos^{2} θ = ∥P_{1}(a)b∥, 0 < θ < \frac{π}{2}. Since γ_{a,θu}(1) = γ_{a,u}(θ) = b, the vector v: = θ u clearly satisfies the requirements. □ Corollary 3.7. With the above notation, the set 𝔐(1) is connected. Proof.

Fix any a ∈ 𝔐(1). Then N_{a}: = {b ∈ 𝔐(1): ab ≠ 0} is an open set which is pathwise connected by (3.5), hence \overline{N_{a}} is also connected. But clearly \overline{N_{a}} = 𝔐(1). □

The set of projections in L(H) that have rank r is known to be connected for every fixed r, 1 ≤ r < ∞, [13]. In order to extend the preceding results, we let 𝔐(r) denote such a set. Suppose that V is a unital Jordan algebra with unit 1 and let p_{1}, ⋯, p_{n} be pairwise orthogonal idempotents with sum 1 . Define V_{i,j}: = {p_{i}V p_{j}}. Then V_{i,j} = V_{j,i} and the vector space direct sum decomposition, called the joint Peirce decomposition relative to the family (p_{k}), holds:

V = ⨁_{1≤i≤j≤n} V_{i,j}

Besides we have the following multiplication rules

V_{i,j}V_{k,l} = 0 if {i, j} ∩ {k, l} = ∅,  V_{i,j}V_{j,k} ⊂ V_{i,k} (pairwise distinct i, j, k),
V_{i,j}V_{i,j} ⊂ V_{i,i} + V_{j,j} (all i, j). (13)

Furthermore we have

(14) {V_{i,j}V_{j,k}V_{k,i}} ⊂ V_{i,i} (all i, j, k), {V_{i,j}V_{j,k}V_{i,j}} = 0 (pairwise distinct i, j, k).

Our goal now is to study V [a, b], where a, b are projections in V that have the same finite rank. To simplify the notation we set W: = V [a, b]. By ([15], lemma 2.5), if V is a topologically simple Jordan JBW-algebra and a, b are two projections in V with the same finite rank, then dim V [a, b] < ∞.

Let e be the unit of W. Since f = (a + b − e)^{2} is a positive selfadjoint element, it has a spectral resolution in W. Let it be

(a + b − e)^{2} = Σ_{1}^{σ}λ_{j}e_{j} (15)

where the number of summands is finite, λ_{j} ≥ 0 and the e_{j} are non zero pairwise orthogonal projections in W. We can also assume that λ_{j} are pairwise distinct and that e = Σ_{1}^{σ}e_{j} though the e_{j} may then fail to be minimal in W. Let

W = ⨁_{i,j} W_{ij},  a = Σ_{i,j}a_{i,j},  b = Σ_{i,j}b_{i,j} (16)

respectively be the Peirce decompositions of W, a and b relative to the complete orthogonal system (e_{j}). Then ([15], lemma 2.2) we have W_{i,j} = {0} for all i ≠ j.

We set W^{j}: = W_{j,j}, a_{j}: = a_{j,j} and b_{j}: = b_{j,j}, (1 ≤ j ≤ σ) to shorten the notation. The decompositions in (16) now read

W = ⨁_{j} W^{j},  a = Σ_{j}a_{j},  b = Σ_{j}b_{j} (17)

Since W = V [a, b] is special, there exists a Jordan^{∗}-isomorphism ω: W → 𝔚 ⊂ 𝔄^{J} for some associative algebra 𝔄. Fix any such isomorphism. Then the elements A := ω(a), B := ω(b) and F := ω(f) satisfy

A ∘ F = AF = ABA = F A = F ∘ A. (18)

These results have been established in [15] in the context of JH-algebras, but a careful reading reveals that no essential use of the scalar product in V has been made. Proposition 3.8. In the decomposition in (17) we have for j = 1, 2, ⋯, σ: (i) The a_{j} are pairwise orthogonal projections and so are the b_{j}. (ii) W^{j} = V [a_{j}, b_{j}], that is, W^{j} is the subalgebra generated in W by a_{j}, b_{j}. (iii) P (a_{j})b_{j} = λ_{j}a_{j} and P (b_{j})a_{j} = λ_{j}b_{j}, where the λ_{j} are as in (15). Proof.

By (13) W^{j}W^{j} ⊂ W^{j} and W^{j}W^{k} = {0} for j ≠ k, hence from a^{2} = a we get

a^{2} = (Σa_{j})^{2} = Σa_{j}^{2}, a = Σa_{k}

Since the W^{k} are direct summands in W we get a_{k}^{2} = a_{k} and similarly b_{k}^{2} = b_{k}.

As a, b and e are elements of ⨁ V_{k} we have V [a, b] ⊂ ⨁ V_{k}. Clearly V_{k} ⊂ W^{k} since a_{k}, b_{k} ∈ W^{k} and so

W = V [a, b] ⊂ ⨁V_{k} ⊂ ⨁W^{k} = W.

Therefore V_{k} = W^{k} since the sum is direct.

To establish the relations in the last assertion, we set

f_{j} := (a_{j} + b_{j} − e_{j})^{2}, (1 ≤ j ≤ σ).

and note that f_{j} ∈ W^{j}. The orthogonality of the W^{j} and (15) yield

f = (a + b − e)^{2} = (Σa_{j} + b_{j} − e_{j})^{2} = Σ(a_{j} + b_{j} − e_{j})^{2} = Σf_{j}
f = (a + b − e)^{2} = Σλ_{j}e_{j}

hence f_{j} = λ_{j}e_{j}, (1 ≤ j ≤ σ). Since W is a special we can transfer the above relations via the Jordan isomorphism ω: W → 𝔚 ⊂ 𝔄^{J}. The relations in (17) via ω yield

AF = A ∘ F = ω(af) = ω(Σλ_{k}a_{k}e_{k}) = ω(Σλ_{k}a_{k}) = Σλ_{k}A_{k},
ABA = ω(aba) = ω(Σa_{k}b_{k}a_{k}) = ΣA_{k}B_{k}A_{k}

which via ω^{−1} gives P (a_{k})b_{k} = λ_{k}a_{k} because the W^{k} are direct summands. Similarly we can prove P (b_{k})a_{k} = μ_{k}a_{k} with μ_{k} = λ_{k}. □

Due to P (a_{k})b_{k} = λ_{k}a_{k} the spectral values λ_{k} satisfy 0 ≤ λ_{k} ≤ 1 and we have the three following possibilities:

Case I: λ_{k} = 0. This can not occur for more than one index, say k = 0. Then P (a_{0})b_{0} = 0 and P (b_{0})a_{0} = 0, hence a_{0} and b_{0} are orthogonal. We shall see below that in this case ranka_{0} = rankb_{0}, (say n_{0}). Thus W^{0} is isomorphic to the space of the diagonal matrices

W ≈ ℝ ( \frac{1}{0} ) ⊕ ℝ ( \frac{0}{1} )

with the usual Jordan matrix operations. Here 1 is the n_{0} × n_{0} unit matrix.

Case II: λ_{k} = 1. This can not occur for more than one index (say k = 1). Then P (a_{1})b_{1} = a_{1} and P (b_{1})a_{1} = b_{1}, which means that a_{1} = b_{1}, hence ranka_{1} = rankb_{1} (say n_{1}) and W^{1} is isomorphic to the space of the diagonal matrices

W^{1} ≈ ℝ ( \begin{matrix}1 \\ & . & . & . \\ & & & & 1 \end{matrix} )

Case III: 0 < λ_{k} < 1. This may occur for several indices k (the corresponding λ being distinct). Then proposition (3.2) applies, hence W^{k} is Jordan isomorphic to Sym (ℝ, 2) via the isomorphism in (3.2). It is now clear that a_{k} and b_{k} are minimal in W^{k}. Since different W^{k} are orthogonal, a_{k} and b_{k} are also minimal in W, that is ranka_{k} = rankb_{k} = 1. Since by assumption a and b had the same rank, we can now conclude that ranka_{0} = rankb_{0} as announced earlier. We can now summarize the discussion in the following Theorem 3.9. Let V be a topologically simple Jordan JBW-algebra and let a, b be two projections in V that have the same finite rank. If W^{0}, W^{1} and W^{k} are the algebras described above, then V [a, b] is Jordan isomorphic to the finite orthogonal direct sum

V [a, b] = W^{0} ⊕ W^{1} ⊕ ⨁_{0≠k≠1} W^{k} (19)

Given a, b in 𝔐(r) we show that it is possible to connect a with b by a geodesic. Lemma 3.10. Let the algebra V, the projections a, b and the decompositions a = Σa_{k} and b = Σb_{k} be as in (17). Then P_{1}(a)b = aba = Σa_{k}b_{k}a_{k} = ΣP_{1}^{k}(a_{k})b_{k}. Proof.

It follows from the facts that W is special and the W^{k} are pairwise orthogonal. Lemma 3.11. Let the algebra V, the projections a, b and the decompositions

a = a_{0} + a_{1} + Σ_{0≠k≠1}a_{k}, b = b_{0} + b_{1} + Σ_{0≠k≠1}b_{k}

be given by (19). If P_{1}(a)b is invertible in the algebra W_{1}(a), then a_{0} = b_{0} = 0, a_{1} = b_{1} and a_{k}b_{k} ≠ 0 for 0 ≠ k ≠ 1. Proof.

It follows directly from the properties of the algebras W^{0}, W^{1} and W^{k} that were established in discussion in (3.9) and the invertibility of P_{1}(a)b. □ Theorem 3.12. Let V = L(H) and let a, b be two projections in V with the same finite rank r. Assume that P_{1}(a)b is invertible in the unital algebra V_{1}(a). Then there is a geodesic that joins a with b in 𝔐(r). Proof.

We may assume a ≠ b. Consider the algebra W: = V [a, b] and the decompositions in (19). By (3.11) the invertibility of P_{1}(a)b in W_{1}(a) yields

a_{0} = b_{0} = 0,  a_{1} = b_{1},  rank(a_{k}) = rank(b_{k}) = 1 for 0 ≠ k ≠ 1

Thus W^{0} = {0} in our case. Let us define γ_{1} : ℝ → W^{1} to be the constant curve γ_{1}(t): = a_{1} = b_{1}, and let r_{1} : = ranka_{1} = rankb_{1}. Clearly γ_{1} is a geodesic in the manifold 𝔐(r_{1}) of the projections in W^{1} that have fixed rank r_{1}.

For 0 ≠ k ≠ 1 the projections a_{k} and b_{k} are non orthogonal and minimal in W^{k}. Hence by (3.5) there is a geodesic, say γ_{k}, that joins a_{k} with b_{k} in 𝔐^{k}(1), the manifold of minimal projections in W^{k}. This curve is of the form

γ_{k}(t) = γ_{a_{k},u_{k}}(t) = [ exp tG(a_{k}, u_{k})] a_{k}, t ∈ ℝ,

where G(a_{k}, u_{k}) := 2(a_{k}□u_{k} − u_{k}□a_{k}) for a tangent vector u_{k} ∈ W_{1/2}^{k}(a_{k}) that is determined by the uniqueness properties established in (3.6). In particular b_{k} = γ_{a_{k},u_{k}}(1). We claim that

γ(t) := γ_{1}(t) + Σ_{k≠1}γ_{k}(t), t ∈ ℝ

is a geodesic that joins a with b in 𝔐(r).

By construction we have γ_{k}(ℝ) ⊂ W^{k}. But these subalgebras are pairwise orthogonal, hence γ(t) is a projection of rank r = r_{1} + Σ_{k≠1}r_{k} for all t ∈ ℝ, that is, γ is a curve in 𝔐(r) and obviously γ(0) = a, γ(1) = b. It remains to show that γ is a geodesic, which amounts to saying that γ is of the form

γ(t) = [ exp tG(a, u)] a, t ∈ ℝ, (20)

for some tangent vector u ∈ W_{1/2}(a), and it is almost clear that u := u_{1} +Σ_{k≠1}u_{k} will do. Indeed, the orthogonality of the W^{k} and the expression of the Peirce projectors P_{1/2}^{k}(a_{k}) and P_{1/2}(a) for special Jordan algebras easily yield the inclusions W_{1/2}^{k}(a_{k}) ⊂ W_{1/2}(a) and so

u = u_{1} + Σ_{k≠1}u_{k} ∈ W_{1/2}^{1}(a_{1}) ⊕ ⨁_{k≠1} W_{1/2}^{k}(a_{k}) ⊂ W_{1/2}(a)

Still we have to check that the equality in (20) holds. To do this, notice that G(W^{j}, W^{k})(W) = {0} for j ≠ k, which is an immediate consequence of (6), the orthogonality properties of the W^{j} and W = ⊕W^{j}. As a consequence G(a, u) a = ΣG(a_{j}, u_{j}) a_{j} = Σw_{j} where w_{j}: = G(a_{j}, u_{j}) a_{j} ∈ W^{j}. Then

G(a, u)^{2} a = G(a, u) G(a, u) a = G(a, u)Σw_{j} =
Σ_{j}G(a_{j}, u_{j}) Σ_{k}w_{k} = Σ_{j}G(a_{j}, u_{j}) w_{j} = Σ_{j}G(a_{j}, u_{j})^{2}a_{j}

and by induction G(a, u)^{n} a = Σ_{j}G(a_{j}, u_{j})^{n} a_{j} for all n ∈ ℕ, hence

[exp tG(a, u)] a = Σ_{j}[exp tG(a_{j}u_{j})] a_{j}

which completes the proof. □

Remark 3.13. The geodesic constructed in (3.12) satisfies certain normalizing conditions. Indeed, the pair (a, b) determines uniquely (up to order) pairs (a_{k}, b_{k}) via the spectral resolution of (a + b − e)^{2} in the unital algebra W [a, b]. In turn, these (a_{k}, b_{k}) determine in a unique way tangent vectors u_{k} ∈ W_{1/2}(a_{k}) such that b_{k} = γ_{a_{k},u_{k}}(1) for 1 ≤ k ≤ r. Finally u = Σ_{k≠1}u_{k}. These properties single out the curve γ in the class of geodesics that connect a with b.

4. Geodesics are minimizing curves.

Throughout this section U stands for the algebra L(H) and V denotes its selfadjoint part. Our next task will be to show that the geodesic γ_{a,u} joining a with b in 𝔐(r) is a minimizing curve. That will require some calculus. Let a be fixed in 𝔐(r), and let | · | denote the Levi norm in V_{1/2}(a) (see 1.2). Notation 4.1. We set N_{a}: = {P_{1}(a)v ∈ 𝔐(r): v is invertible in V_{1}(a) }. Clearly N_{a} is an open neighbourhood of a in 𝔐(r) and {N_{a}: a ∈ 𝔐(r)} is an open cover of 𝔐(r).

By B_{a}: = \{x ∈ V_{1/2}(a): ∥x∥ < \frac{π}{2}\} we denote the open ball in V_{1/2}(a) of radius \frac{π}{2} centered at the origin. Using the odd functional calculus for the JB^{∗}-triple V_{1/2}(a) (see [10]) one can define a mapping ρ: B_{a} → V_{1/2}(a) by

u ↦ ρ(u): = tan u, u ∈ B_{a}. (21)

Then ρ is a real analytic diffeomorphism of B_{a} onto V_{1/2}(a) whose inverse, also defined by the odd functional calculus, is

u ↦ σ(u): = arctan u, u ∈ V_{1/2}(a).

Definition 4.2. We define Φ_{a}: N_{a} → V_{1/2}(a) and Ψ_{a}: V_{1/2}(a) → V by

(22)  Φ_{a}(v): = 2 (P_{1}(a)v)^{−1}P_{1/2}(a)v,
(23)  Ψ_{a}(u): = [exp G(a, σ(u))] a,

Lemma 4.3. With the above notation, Φ_{a} and Ψ_{a} are real analytic V -valued functions. Furthermore Φ_{a}(N_{a}) ⊂ V_{1/2}(a) and Ψ_{a}(V_{1/2}(a)) ⊂ N_{a}. Proof.

For v ∈ N_{a}, P_{1}(a)v is invertible in V_{1}(a). Hence the mapping v ↦ (P_{1}(a)v)^{−1} is well defined and real analytic in N_{a}. Clearly v ↦ P_{1/2}(a)v is real analytic, hence the product of these two functions, that is, Φ_{a} is also real analytic and by the Peirce multiplication rules Φ_{a}(N_{a}) ⊂ V_{1/2}(a).

As said before u ↦ tan(u) is a real analytic V -valued function, and so is u ↦ G(a, tan(u))a since G is a continuous real bilinear mapping. The exponential u ↦ exp G(a, tan(u)) is an operator-valued real analytic function, hence by evaluating at a we get Ψ_{a}, a real analytic function. Let u ∈ V_{1/2}(a). We have the decompositions

a = a_{0} + Σ_{k}a_{k}, u = Σ_{k}ξ_{k}u_{k},

given by (11) with the properties in the statement of theorem (2.5). The odd functional calculus and orthogonality gives

arctan u = Σ_{k}θ_{k}u_{k}, where θ_{k} := arctan ξ_{k}.

Hence G(a, σ(u)) = Σ_{k}θ_{k}G(a_{k}, u_{k}). Again using the orthogonality properties G(a_{k}, u_{k})V [a_{j}, u_{j}] = 0 for k ≠ j we see (recalling the proof of 2.3) that

Ψ_{a}(u) = [exp G(a, σ(u))] a = Σ_{k}[exp θ_{k}G(a_{k}, u_{k})] a_{k} =
Σ_{k}(cos^{2} θ_{k}) a_{k} + Σ_{k}(\frac{1}{2} sin 2θ_{k})u_{k} + Σ_{k}(sin^{2} θ_{k}) u_{k}^{(2)}.

(24)

An inspection of this formula shows that P_{1}(a)Ψ_{a}(u) = Σ_{k}(cos^{2} θ_{k})a_{k}. Since arctan u ∈ B_{a}, we have that max θ_{k} = ∥ arctan u∥ < \frac{π}{2}. In particular, P_{1}(a)Ψ_{a}(u) is invertible in V_{1}(a), that is Ψ_{a}(u) ∈ N_{a}. Therefore Ψ_{a}(V_{1/2}(a)) ⊂ N_{a} which completes the proof. □

Proposition 4.4. With the above notation, Φ_{a}: N_{a} → V_{1/2}(a) is a real analytic diffeomorphism of N_{a} onto V_{1/2}(a) whose inverse is Ψ_{a}. Proof.

First we show that Φ_{a} is invertible in a suitable neighbourhood W ⊂ N_{a} of a. Let us use the following self-explanatory notation

Φ_{a}(v) = 2 (P_{1}(a)v)^{−1} P_{1/2}(a)(v): = 2f(v) g(v).

Note that f(a) = a and g(a) = P_{1/2}(a)(a) = 0. Thus for h ∈ V_{1/2}(a) we have

Φ′_{a}(a)h = 2 (f′(a)h) g(a) + 2f(a) g′(a)h = 2aP_{1/2}(a)h = h,

that is Φ′_{a}(a) = Id which by the inverse mapping theorem proves the first claim. Let v ∈ V_{1/2}(a) and u ∈ B_{a} be related by v = tan(u). A glance at (24) shows

P_{1}(a)Ψ_{a}(u) = Σ_{k}(cos^{2} θ_{k}) a_{k} and P_{1/2}(a)Ψ_{a}(u) = Σ_{k}(\frac{1}{2} sin 2θ_{k}) u_{k}.

Therefore since u_{k} ∈ V_{1/2}(a_{k}),

Φ_{a}Ψ_{a}(u) = 2 (P_{1}(a)Ψ_{a}(u))^{−1} P_{1/2}(a)Ψ_{a}(u) = 2 Σ_{k}(tan θ_{k}) a_{k}u_{k} = Σ_{k}ξ_{k}u_{k} = u.

Hence Φ_{a}Ψ_{a} = Id. In particular Ψ_{a} is the right-inverse of Φ_{a}, and the inverse of Φ_{a} at least in W. By (4.3), the mappings Φ_{a}Ψ: V_{1/2}(a) → V_{1/2}(a) and Ψ_{a}Φ_{a}: N_{a} → N_{a} are well defined and analytic in their respective domains. By (3.12) any point in N_{a} can be joined with a by a geodesic that is contained in N_{a}, hence N_{a} is an open connected set. Since Ψ_{a}Φ_{a} = Id in W, we have Ψ_{a}Φ_{a} = Id in N_{a} by the identity principle. This completes the proof. □ Proposition 4.5. The family of charts {(N_{a}, Φ_{a}): a ∈ 𝔐(r)} is an atlas which defines the manifold 𝔐(r). Proof.

It is easy to check that the above family is a real analytic atlas whose manifold structure is denoted by 𝔐(r)′. To see that 𝔐(r)′ is the same as 𝔐(r), recall that

\begin{matrix}f: U ⊂ V_{1/2}(a) → 𝔐(r) & Φ_{a}: 𝔐(r)′ → V_{1/2}(a) \\ u ↦ f(u): = [exp G(a, u)] a & v ↦ Φ_{a}(v): = 2 (P_{1}(a)v)^{−1} P_{1/2}(a)v \end{matrix}

are local charts of 𝔐(r) and 𝔐(r)′ at the point a. The composite map F := Φ_{a} ∘ f can be written in the form

F (u) = Φ_{a}[f(u)] = 2 (P_{1}(a)f(u))^{−1} P_{1/2}(a)f(u) = G(u) H(u)

with self-explanatory notation. Then G(0) = 2a, H(0) = P_{1/2}(a)a = 0, H′(0) = P_{1/2}(a). Therefore, for h ∈ V_{1/2}(a) we have

F′(0)h = (G′(0) h)H(0) + G(0)H′(0)h = G(0)H′(0)h = 2a P_{1/2}(a)h = h.

Thus F′(0) = Id. The remaining part of the proof is similar. □

We are in the position to prove that geodesics in 𝔐(r) are minimizing curves. For that we consider 𝔐(r) as defined by the atlas {(N_{a}, Φ_{a}): a ∈ 𝔐(r)}. Theorem 4.6. Let 𝔐(r) be the manifold of projections in V = L(H) that have fixed finite rank r. Fix a ∈ 𝔐(r) and let N_{a} be defined as in (4.1) Then for every b in N_{a}, the geodesic joining a with b is a minimizing curve for the Riemann distance in N_{a}.

Proof.

We may assume a ≠ b. The diffeomorphisms B_{a} ⟶^{ρ} V_{1/2}(a) ⟶^{Ψ_{a}} N_{a} give a unique pair (u, v) ∈ B_{a} × V_{1/2}(a) such that

v = tan(u) Ψ_{a}(v) = b.

There is a unique normalized geodesic γ_{a,u} that joins a with b in the manifold 𝔐(r) and has initial velocity u = γ̇_{a,u}(0) ∈ B_{a}. In particular we have

b = Ψ_{a}(ρ(u)) = γ_{a,u}(1) = [exp G(a, u)] a

and the exponential mapping exp: B_{a} → 𝔐(r) is a homeomorphism of B_{a} onto the open set N_{a} in 𝔐(r). This will allow us to apply the Gauss lemma ([12] 1.9). For that purpose, we show that γ_{a,u}(t) belongs to N_{a} for all t ∈ [0, 1]. Indeed, the segment [0, 1] u is contained in B_{a}, hence its image by Ψ_{a} ∘ ρ lies in the set N_{a}. We shall now see that

Ψ_{a}[ρ(tu)] = γ_{a,u}(t), t ∈ [0, 1].

Let t ∈ [0, 1] and set v_{t}: = tan(tu). The odd functional calculus gives

Ψ_{a}(v_{t}) = [exp G(a, tu)] a = [exp t G(a, u)] a = γ_{a,u}(t)

as we wanted to see. For the Riemann connection ∇ in 𝔐(r), the radial geodesics are minimizing curves (by the Gauss lemma). Hence it suffices to see that γ_{a,u} is a radial geodesic, which is a consequence of the fact γ_{a,u}[0, 1] ⊂ N_{a}. □

It is now reasonable to ask what can be said about the neighbourhood N_{a}. We refer to O_{a}: = {b ∈ 𝔐(r): P_{1}(a)b is not invertible in V_{1}(a)} as the antipodal set of a. Clearly O_{a} is a closed subset of 𝔐(r). Proposition 4.7. Let 𝔐(r) be the manifold of all projections in V = L(H) that have a given finite rank r. Then for any a ∈ 𝔐(r) the antipodal set of a has empty interior. Proof.

Let a ∈ 𝔐(r) and set K: = a(H) ⊂ H. Note that dimK = ranka = r < ∞. The operators in V_{1}(a) = aV a can be viewed as operators in L(K), therefore the determinant function is defined in V_{1}(a) and an element z ∈ V_{1}(a) is invertible if and only if det(z) ≠ 0. The function b ↦ det(P_{1}(a)b) is real analytic on 𝔐(r). If O_{a} has non empty interior, then det(P_{1}(a)b) vanishes in a non void open subset of 𝔐(r), which is connected, therefore by the identity principle the determinant function would be identically null which is a contradiction. □

We let 𝔖(r) denote the subgroup of Aut(U) generated by the set of Peirce reflections σ_{p}, where p ∈ 𝔐(r). Each σ_{p} preserves 𝔐(r) and induces a real analytic symmetry of this manifold. Proposition 4.8. Let V = L(H). Then 𝔐(r) is homogeneous under the action of any of the groups Aut^{∘}(V) and 𝔖(r). Proof.

Let a, b be any pair of points in 𝔐(r) with a ≠ b. If b ∈ N_{a} then by (3.12) there is a geodesic γ_{a,u}(t) = [exp tG(a, u)] a that joins a with b in 𝔐(r). But g(t): = [exp tG(a, u)] is an element of Aut^{∘}(V) for all t ∈ ℝ. Now consider the case b ∉ N_{a}. By (4.7) the antipodal set of a has empty interior, hence W ∩ N_{a} ≠ ∅ for every neighbourhood W of b in 𝔐(r). If c ∈ W ∩ N_{a} then we can connect a with c and c with b by geodesics in 𝔐(r) hence we connect a with b by a curve in 𝔐(r). This completes the proof for Aut^{∘}(V).

Consider a pair of points a, b in 𝔐(r) such that b ∈ N_{a}. Then it is easy to find a symmetry σ_{p} that exchanges a with b. Namely, there is a geodesic that connects a with b in 𝔐(r), say γ: t ↦ γ(t), t ∈ ℝ. Thus a = γ(0) and b = γ(1). Define the geodesic middle point of the pair (a, b) as c: = γ(\frac{1}{2}), and let σ_{c} be the Peirce reflection with center at c. Clearly σ_{c} preserves the curve γ and exchanges a with b.

Now let a, b be arbitrary in 𝔐(r). Since 𝔐(r) is a connected locally path-wise connected topological space, it is globally path-wise connected. Thus there is a curve Γ: t ↦ Γ(t), t ∈ [0, 1], that joins a with b in 𝔐(r), and a standard compactness argument shows that there is a finite sequence of points {x_{1}, ⋯, x_{r}} in Γ such that x_{1} = a, x_{r} = b and each consecutive pair of the x_{k} can be joined by a geodesic in 𝔐(r). For each pair (x_{k}, x_{k+1}) consider the geodesic middle point c_{k} and the corresponding symmetry exchanging x_{k} with x_{k+1}. Then the composite σ_{1} ∘ ⋯ ∘ σ_{k} lies in 𝔖(r) and exchanges a with b. □

Corollary 4.9. Let a, b ∈ 𝔐(r). Then b ∈ O_{a} if and only if a ∈ O_{b}.

Proof.

Let σ be a symmetry that exchanges a with b. The relation b ∈ O_{a} is equivalent to P_{1}(a)b is not invertible in V_{1}(a), which applying σ is converted into P_{1}(b)a is not invertible in V_{1}(b) that is a ∈ O_{b}. □

We are now in the position to compute the Riemann distance in 𝔐(r).

Theorem 4.10. Let 𝔐(r) be the manifold of all projections in V = L(H) that have a given finite rank r. If a, b are points in 𝔐(r) and γ_{a,u}(t) is the normalized geodesic connecting a with b in 𝔐(r) then the Riemann distance between them is

d(a, b) = (Σ_{1}^{r}θ_{k}^{2})^{\frac{1}{2}}

where θ_{k} = cos^{−1} (∥P_{1}(a_{k})b_{k}∥^{\frac{1}{2}} ) and ∥ · ∥ stands for the usual operator norm.

Proof.

Discard the trivial case a = b. Consider first the case b ∈ N_{a}. By (3.12, 3.13) we have b = γ_{a,u}(1) for some tangent vector u = Σu_{k} where u_{k} = θ_{k} v_{k}, 0 < θ_{k} < \frac{π}{2}, (1 ≤ k ≤ r), and the v_{k} are pairwise orthogonal (in the JB^{∗}-triple sense) minimal tripotents in W_{1/2}(a_{k}). Hence the v_{k} are pairwise orthogonal in the Levi sense (see [1] prop. 9.12 and 9.13) and so, if | · |_{a} denotes the Levi norm in the tangent space V_{1/2}(a), we have |u|_{a}^{2} = Σ_{1}^{r}θ_{k}^{2} (recall that minimal tripotents satisfy ∥v∥ = |v|_{a} = 1). Therefore since the Levi form is Aut^{∘}(V)-invariant and Aut^{∘}(V) is transitive in 𝔐(r),

|γ̇_{a,u}(t)|_{γ_{a,u}(t)} = |γ̇_{a,u}(0)|_{a} = |u|_{a} = (Σ_{1}^{r}θ_{k}^{2})^{\frac{1}{2}}

for all t ∈ ℝ, hence

d(a, b) = ∫_{0}^{1} |γ̇_{a,u}(t)|_{γ_{a,u}(t)}dt = |u|_{a} = (Σ_{1}^{r}θ_{k}^{2})^{\frac{1}{2}} .

Here θ_{k} = cos^{−1} (∥P_{1}(a_{k})b_{k}∥)^{\frac{1}{2}} and ∥ · ∥ is the JB^{∗}-triple (that is, the usual operator) norm. In the case b ∈ O_{a}, consider a sequence (b_{j})_{j∈N_{a}} with b_{j} → b in 𝔐(r) and ab_{j} ≠ 0 for all j. Applying the above to each j and taking the limit we get the result. □

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