THE MANIFOLD OF FINITE RANK PROJECTIONS IN THE SPACE L(H).

José M. Isidro †

Universidad de Santiago de Compostela

February, 2000

Abstract. Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators in H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. To do it we take the Jordan-Banach triple systems (or JB^{∗}-triple) approach which allows us to define the Levi-Civita connection on M by using elementary tools. We identify the geodesics and the Riemann distance and establish some properties of M.

0 Introduction

In this paper we are concerned with the differential geometry of the infinite-dimensional Grassmann manifold M of all projections in Z: = L(H), the space of bounded linear operators z: H → H in a complex Hilbert space H. Grassmann manifolds are a classical object in Differential Geometry and in recent years several authors have considered them in the Banach space setting. Besides the Grassmann structure, a Riemann and a Kähler structure has sometimes been defined even in the infinite-dimensional setting. Let us recall some aspects of the history of the topic that are relevant for our purpose.

The study of the manifold of minimal projections in a finite-dimensional simple formally real Jordan algebra was made by U. Hirzebruch in [4], who proved that such a manifold is a compact symmetric Riemann space of rank 1, and that every such a space arises in this way. Later on, Nomura in [13, 14] established similar results for the manifold of fixed finite rank projections in a topologically simple real Jordan-Hilbert algebra. On the other hand, the Grassmann manifold M of all projections in the space Z: = L(H) of bounded linear operators has been discussed by Kaup in [7] and [10]. See also [16,17,18] for related results. It is therefore reasonable to ask whether a Riemann structure can always be defined in M and how does it behave when it exists. It is known that M has several connected components M_{r} ⊂ M each of which consists of the projections in L(H) that have a fixed rank r, 1 ≤ r ≤ ∞. We prove that M_{r} admits a Riemann structure if and only if r < ∞ establishing a distinction between the finite and the infinite dimensional cases. We then assume r < ∞ and proceed to discuss the behaviour of the Riemann manifold M_{r}. One of the novelties is that we take JB^{∗}-triple systems approach instead of the Jordan-algebra approach of [4] and [13]. As noted in [1] and [5], within this context the algebraic structure of JB^{∗}-triple acts as a substitute for the Jordan algebra structure and provides a local scalar product known as the Levi form [10]. Although L(H) is not a Hilbert space, the JB^{∗}-triple approach and the use of the Levi form allows us to define a torsion-free affine connection ∇ on

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 48 G 20, 72 H 51.

Key words and phrases. JB*-triple systems, Bounded operators, Riemann manifolds.

†Supported by Comisión Hispano-Húngara de Cooperación Científica y Tecnológica

2 JOSÉ M. ISIDRO †

M_{r} that is invariant under the group Aut^{∘}(Z) of all surjective linear isometries of L(H). We integrate the equation of the geodesics and define an Aut^{∘}(Z)-invariant Riemann metric on M_{r} with respect to which ∇ is a Levi-Civita connection. We prove that any two distinct points in M_{r} can be joined by a geodesic which (except for the case of a pair of antipodal points) is uniquely determined and is a minimizing curve for the Riemann distance, that is also computed. We prove that M_{r} is a symmetric manifold on which Aut^{∘}(Z) acts transitively as a group of isometries.

1 JB^{∗}-triples and tripotents.

For a complex Banach space Z, denote by L(Z) the Banach algebra of all bounded linear operators on Z. A complex Banach space Z with a continuous mapping (a, b, c) ↦ {abc} from Z × Z × Z to Z is called a JB*-triple if the following conditions are satisfied for all a, b, c, d ∈ Z, where the operator a□b ∈ L(Z) is defined by z ↦ {abz} and [ , ] is the commutator product:

1 {abc} is symmetric complex linear in a, c and conjugate linear in b.

2 [a□b, c□d] = {abc}□d − c□{dab}.

3 a□a is hermitian and has spectrum ≥ 0.

4 ∥{aaa}∥ = ∥a∥^{3}.

If a complex vector space Z admits a JB*-triple structure, then the norm and the triple product determine each other. A derivation of a JB*-triple Z is an element δ ∈ L(Z) such that δ{zzz} = {(δz)zz} + {z(δz)z} + {zz(δz)} and an automorphism is a bijection ϕ ∈ L(Z) such that ϕ{zzz} = {(ϕz)(ϕz)(ϕz)} for z ∈ Z. The latter occurs if and only if ϕ is a surjective linear isometry of Z. The group Aut(Z) of automorphisms of Z is a real Banach-Lie group whose Banach-Lie algebra is the set of derivations of Z . The connected component of the identity in Aut(Z) is denoted by Aut^{∘}(Z). Two elements x, y ∈ Z are orthogonal if x□y = 0. An element e ∈ Z is called a tripotent if {eee} = e. The set Tri(Z) of tripotents is endowed with the induced topology of Z. If e ∈ Tri(Z), then e□e ∈ L(Z) has the eigenvalues 0, \frac{1}{2}, 1 and we have the topological direct sum decomposition

Z = Z_{1}(e) ⊕ Z_{1/2}(e) ⊕ Z_{0}(e)

called the Peirce decomposition of Z. Here Z_{k}(e) is the k- eigenspace and the Peirce projections are

P_{1}(e) = Q^{2}(e),  P_{1/2}(e) = 2(e□e − Q^{2}(e)),  P_{0}(e) = Id − 2e□e + Q^{2}(e),

where Q(e)z = {eze} for z ∈ Z. We will use the Peirce rules {Z_{i}(e) Z_{j}(e) Z_{k}(e)} ⊂ Z_{i−j+k}(e) where Z_{l}(e) = {0} for l ≠ 0, 1/2, 1. We note that Z_{1}(e) is a complex unital JB*-algebra in the product a ∘ b: = {aeb} and involution a^{#}: = {eae}. Let

A(e): = {z ∈ Z_{1}(e) : z^{#} = z}.

Then we have Z_{1}(e) = A(e) ⊕ iA(e). The Peirce spaces of Z with respect to a an orthogonal family of tripotents E = (e_{i})_{i∈I} are defined by

Z_{ii}: = Z_{1}(e_{i})
Z_{ij}: = Z_{1/2}(e_{i}) ∩ Z_{1/2}(e_{j}), i ≠ j
Z_{i0}: = Z_{0i}: = Z_{1/2}(e_{i}) ⋂_{j≠i} Z_{0}(e_{j})
Z_{00}: = ⋂_{i∈I} Z_{0}(e_{i})

The Peirce sum P (E): = ⨁_{i,j∈I} Z_{ij} relative to the family E is direct and we have Z = P (E) whenever E is a finite set. Every E-Peirce space is a JB^{∗}-subtriple of Z and the Peirce rules

{Z_{ij}Z_{jk}Z_{kl}} ⊂ Z_{il}

hold for all i, j, k, l ∈ I.

A tripotent e in a JB^{∗}-triple Z is said to be minimal if P_{1}(e)Z = ℂe, and we let Min(Z) be the set of them. Clearly e = 0 lies in Min(Z) and is an isolated point there. If e ∈ Min(Z) and e ≠ 0 then ∥e∥ = 1 and by the Peirce multiplication rules we have {euv} ∈ Z_{1}(e) = ℂe for all u, v ∈ Z_{1/2}(e). Therefore we can define a sesquilinear form, called the Levi form, ⟨·, ·⟩_{e}: Z_{1/2}(e) × Z_{1/2}(e) → ℂ by

{euv} = ⟨v, u⟩_{e} e, u, v ∈ Z_{1/2}(e).

It is known [10] that ⟨·, ·⟩_{e} is positive definite hence it defines a scalar product in Z_{1/2}(e) whose norm, called the Levi norm and denoted by | · |_{e}, satisfies

|u|_{e}^{2} ≤ ∥u∥^{2}, u ∈ Z_{1/2}(e)

that is, we have the continuous inclusion (Z_{1/2}(e), ∥ · ∥) ↪ (Z_{1/2}(e), | · |_{e}). To simplify the notation, we shall omit the subindex e in both the Levi form and the Levi norm if no confusion is likely to occur.

JB*-triples include C*-algebras and JB*-algebras. A C*-algebra is a JB*-triple with respect to the triple product 2{abc}: = (ab^{∗}c + cb^{∗}a). Every JB*-algebra with Jordan product (a, b) ↦ a ∘ b and involution a ↦ a^{∗} is a JB*-triple with triple product {abc} = (a∘b^{∗})∘c−(c∘a)∘b^{∗}+(b^{∗}∘c)∘a.

We refer to [8,9,10,12] for the background of JB^{∗}-triples theory.

2 The manifold M of minimal projections

Let Z: = L(H), where H is a complex Hilbert space, and let M ⊂ Tri(Z) denote the set of all projections in Z endowed with its topology as subspace of Z. Fix any non zero projection e_{0} ∈ M and denote by M the connected component of e_{0} in M. Then all elements in M have the same rank as e_{0} and Aut^{∘}(Z) acts transitively on M which is an Aut^{∘}(Z)- invariant real analytic manifold whose tangent space at a point e ∈ M is

T_{e}M = Z_{1/2}(e)_{s},

the selfadjoint part of the \frac{1}{2}-eigenspace of e. If we set k_{u}: = 2(u□e − e□u), then by [1, th. 3.3] a local chart of M in a suitable neighbourhood U of 0 in Z_{1/2}(e)_{s} is given by

u ↦ f(u): = exp k_{u}(e).

Let 𝔇(M) be the Lie algebra of all real analytic vector fields on M, and as in [1], define an affine connection ∇ on M by

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

Then ∇ is a torsion-free Aut^{∘}(Z)-invariant affine connection on M. For each e ∈ M and u ∈ Z_{1/2}(e)_{s} we let γ_{e,u}: ℝ → M denote the curve γ_{e,u}(t): = exp tk_{u}(e). Clearly we have γ_{e,u}(0) = e and γ̇_{e,u}(0) = u ∈ T_{e}M. By [1, th. 2.7], γ_{e,u} is a ∇-geodesic of M. Let us introduce a binary product in Z by x ∘ y: = {xey}. Then (Z, ∘) is a complex Jordan algebra where, as usual, x^{(n)} denotes the n-th power of x in (Z, ∘) for n ∈ ℕ. For u ∈ Z_{1/2}(e), the real Jordan subalgebra of (Z, ∘) generated by the pair (e, u) is denoted by J[e, u] and we have γ_{e,u}(ℝ) ⊂ J[e, u].

To make a more detailed study of the manifold M, we shall assume that e_{0} is minimal. In such a case J[e, u] coincides with the closed real linear span of the set {e, u, u^{(2)}}, in particular dim J[e, u] ≤ 3 and

γ_{e,u}(t) = (cos^{2} tθ) e + (\frac{1}{2θ} sin 2tθ) u + (\frac{1}{θ^{2}} sin^{2} tθ) u^{(2)},  t ∈ ℝ  (2)

for some angle 0 ≤ θ < \frac{π}{2} . If a, b are two distinct minimal projections and they are not orthogonal (that is, if the Peirce projection P_{1}(a)b is invertible in the JB^{∗}-algebra Z_{1}(a)) then there is an unique geodesic γ_{a,u}(t) joining a with b in M. Moreover, due to the minimality of e the tangent space Z_{1/2}(e) ≈ {e}^{⊥} appears naturally endowed with the Levi form ⟨·, ⟩_{e} and it turns out that the Levi norm | · |_{e} and the operator norm ∥ · ∥ are equivalent in Z_{1/2}(e) (see [6, th.5.1]). Thus (Z_{1/2}(e), | · |_{e}) is a Hilbert space and an Aut^{∘}-invariant Riemann structure can be defined in M by

g_{e}(X, Y): = ⟨X_{e}, Y_{e}⟩_{e},  X, Y ∈ 𝔇(M)  (3)

where V_{e} ∈ Z_{1/2}(e) denotes the value taken by the vector field V at the point e ∈ M. By [1] g satisfies

Xg(Y, Z) = g(∇_{X}Y, Z) + g(Y, ∇_{X}Z),  X, Y, Z ∈ 𝔇(M)  (4)

Therefore ∇ is the only Levi-Civita affine connection on M, and the geodesics are minimizing curves for the Riemann distance in M, which is given by the formula

d(a, b) = cos^{−1} (∥P_{1}(a)b∥^{\frac{1}{2}} ) = θ.

M is symmetric Riemann manifold on which Aut^{∘}(Z) acts transitively as a group of isometries and there is a real analytic diffeomorphism of M onto the projective space ℙ(H) over H, endowed with the Fubini-Study metric. We refer to [1,5,6,13] for proofs and background about these facts.

3 The manifold of finite rank projections in L(H).

In what follows we let M and M_{r} be the set of all projections in Z and the set of all projections that have a fixed finite rank r, respectively. If a ∈ M_{r} then a frame for a is any family (a_{1}, ⋯, a_{r}) of pairwise orthognal minimal projections in Z such that a = Σa_{k}. Note that then the a_{k} have the form a_{k} = (·, α_{k})α_{k} where (α_{k}) is an orthonormal family of vectors in the range a(H).

3.1 Proposition. For every projection a ∈ M the following conditions are equivalent: (1) The rank of a is finite.

(2) The Banach space Z_{1/2}(a) is linearly homeomorphic to a Hilbert space.

Proof. Let us choose an orthonormal basis (α_{ı})_{ı∈I} in the range a(H) ⊂ H of a. Then a_{ı}: = (·, α_{ı})α_{ı}, ı ∈ I, is a family of pairwise orthogonal minimal projections that satisfy

a = Σ_{ı∈I}a_{ı}  strong operator convergence in Z  (5)

The space Z_{1/2}(a)_{s} consists of the operators u ∈ Z such that 2{aau} = u and using (5) it is easy to check that u can be represented in the form

u = Σ_{ı∈I}(·, ξ_{ı})α_{ı} + (·, α_{ı})ξ_{ı} strong operator convergence in Z

where ξ_{ı}: = u(α_{ı}) are vectors in H that satisfy ξ_{ı} ∈ a(H)^{⊥}. By (4) each u ∈ Z_{1/2}(a)_{s} is determined by the family (ξ_{ı})_{ı∈I}. To simplify the notation, set K: = a(H)^{⊥} and L: = ℓ_{∞}(I, K) for the Banach space of the families (ξ_{ı})_{ı∈I} ⊂ K with the norm of the supremun ∥(ξ_{ı})∥: = sup_{ı∈I} ∥ξ_{ı}∥. Then the mapping

L → Z_{1/2}(a)_{s}, (ξ_{ı}) ↦ u_{ξ}: = Σ_{ı∈I} [ (·, α_{ı})ξ_{ı} + (·, ξ_{ı})α_{ı}]

is a continuous real linear vector space isomorphism, hence a homeomorphism . Thus if the operator norm in Z_{1/2}(a)_{s} is equivalent to a Hilbert space norm the same must occur with ℓ_{∞}(I, K), hence I must be a finite set which means that a = Σa_{ı} has finite rank. The converse is easy. □

3.2 Lemma. Let a, b ∈ M_{r} with a = Σa_{k} where the (a_{k}) is a frame for a, and let Q(a_{k})b = λ_{k}a_{k}, (k = 1, ⋯, r). If P_{1}(a)b is invertible in the JB^{∗}-algebtra Z_{1}(a), then λ_{k} ≠ 0 for all k. The set of all elements b ∈ M_{r} for which P_{1}(a)b is invertible in Z_{1}(a) is dense in M_{r}.

Proof. Suppose that a_{k} = (·, α_{k})α_{k} and b_{j} = (·, β_{j})β_{j} are frames for a and b respectively. Then for each fixed k we have

Q(a_{k})b = {a_{k}ba_{k}} = (Σ_{j}|(α_{k}, β_{j})|^{2})a_{k} = λ_{k}a_{k}

where λ_{k} ≥ 0. Moreover λ_{k} = 0 if and only if α_{k} ∈ {β_{1}, ⋯, β_{r}}^{⊥} which is equivalent to a_{k} ⊥ b. But in such a case range(a_{k}) ⊂ ker{a_{k}ba_{k}} = kerP_{1}(a)b which contradicts the invertibility of P_{1}(a)b. To simplify the notation set K: = a(H) ⊂ H and note that dim K = rank a = r < ∞ The operators in Z_{1}(a) = aZa can be viewed as operators in L(K), therefore the determinant function is defined in Z_{1}(a) and an element z ∈ Z_{1}(a) is invertible if and only if det (z) ≠ 0. Thus the set of the operators b ∈ Z for which P_{1}(a)b is invertible in Z_{1}(a) is an open dense subset of M_{r}. □

3.3 Lemma. If a, p and q are projections in M_{r} and P_{1/2}(a)p = P_{1/2}(a)q, then p = q.

Proof. Take frames for a, p, q, compute P_{1/2}(a)p = 2(D(a□a) − Q(a)^{2})p and proceed similarly with q. An elementary exercise of linear algebra yields range (p)=range (q), hence p = q. □

Let a ∈ M_{r} and choose any frame (a_{1}, a_{2}, ⋯, a_{r}) for a. As above Z_{1/2}(a)_{s} consists of the operators u = Σ(·, ξ_{k})α_{k} + (·, α_{k})ξ_{k} where ξ_{k}: = u(α_{k}) are vectors in H that satisfy ξ_{k} ∈ a(H)^{⊥}. Write u_{k}: = (·, ξ_{k})α_{k} + (·, α_{k})ξ_{k}. Then we have u = Σu_{k} where the u_{k} are selfadjoint operators in Z = L(H) (in fact u_{k} ∈ Z_{1/2}(a_{k})_{s}) that satisfy

u_{j}□a_{k} = a_{k}□u_{j} = 0,  j ≠ k,  (j, k = 1, 2, ⋯, r)  (6).

The above properties of the a_{k}, u_{k} hold whatever is the frame (a_{1}, a_{2}, ⋯, a_{r}). There are many families in those conditions and we are going to prove that, by making an appropriate choice of the a_{k} (a choice in which the tangent vector u ∈ Z_{1/2}(a) is also involved) we can additionally have

u_{k}□u_{j} = u_{j}□u_{k} = 0,  j ≠ k,  (j, k = 1, 2, ⋯, r)  (7)

This will simplify considerably the calculations in the sequel. We need some material.

3.4 Lemma. With the above notation the set of minimal tripotents in Z_{1/2}(a) is

{ (·, α)ξ + (·, ξ)α : α ∈ a(H), ξ ∈ a(H)^{⊥}, ∥α∥ = 1 = ∥ξ∥ }

Proof. Let x ∈ Z be of the form x = (·, α)ξ +(·, ξ)α where α, ξ ∈ H satisfy the above conditions. It is a matter of routine calculation to see that then 2{aax} = x hence x ∈ Z_{1/2}(a)_{s}. Moreover {xxx} = x so that x is a tripotent and we can easily see that {xZ_{1/2}(a)x} ⊂ ℂx which proves the minimality of x in Z_{1/2}(a). The converse is similar. □

The following result should be compared to [14, prop. 3.4] 3.5 Lemma. Two minimal tripotents x = (·, α)ξ + (·, ξ)α and y = (·, β)η + (·, η)β in Z_{1/2}(a)_{s} are orthogonal if and only if α ⊥ β and ξ ⊥ η. In particular Z_{1/2}(a) has rank r for all a ∈ M

Proof. By [2, p. 18] x and y are orthogonal if and only if the conditions xy^{∗} = 0 = y^{∗}x hold. Now it is elementary to complete the proof of the first statement. For the second part, let (u_{ı})_{ı∈I} be a family of pairwise minimal orthogonal tripotents in Z_{1/2}(a). Then u_{ı} = (·, α_{ı})ξ_{ı} + (·, ξ_{ı})α_{ı} where (α_{ı}) ⊂ a(H) and ξ_{ı} ⊂ a(H)^{⊥} are orthonormal families of vectors in H. In particular a_{ı}: = (·, α_{ı})α_{ı} is a family of pairwise orthogonal projections with Σa_{ı} ≤ a. Since rank(a)=r, we have cardinal (I) ≤ r. The converse is easy. □

Let a ∈ M be a fixed projection and take any tangent vector u ∈ Z_{1/2}(a)_{s} to M at a. By lemma 3.2 Z_{1/2}(a) has finite rank, hence [9, cor 4.5] u has a spectral decomposition in the JB^{∗}-triple Z_{1/2}(a) of the form

u = ρ_{1}u_{1} + ⋯ + ρ_{s}u_{s},  0 ≤ ρ_{1} ≤ ⋯ ≤ ρ_{s} = ∥u∥,  1 ≤ s ≤ r  (8)

where the u_{k} are pairwise orthogonal minimal tripotents in Z_{1/2}(a). Therefore

u_{k} = (·, α_{k})ξ_{k} + (·, ξ_{k})α_{k}, α_{k} ∈ a(H), ξ_{k} ∈ a(H)^{⊥},
∥α_{k}∥ = 1 = ∥ξ_{k}∥, α_{j} ⊥ α_{k}, ξ_{j} ⊥ ξ_{k}, j ≠ k

Then a_{k}: = (·, α_{k})α_{k} are pairwise orthogonal minimal projections in Z and Σa_{k} ≤ a. In case s < r, which occurs if some of the ρ_{k} = 0, we pick additional minimal orthogonal projections a_{s+1}, ⋯, a_{r} so as to have a = Σa_{k}. For the family (a_{1}, ⋯, a_{r}) so constructed, called a frame associated to the pair (a, u), both properties (6) and (7) hold. Remark that this frame needs not be unique, it depends on a and on u as well, and it is invariant under the group Aut^{∘}(Z). In fact some more properties are valid now.

In accordance with section §1, each pair (a_{k}, u_{k}) gives rise to a real Jordan algebra J_{k}: = J[a_{k}, u_{k}] with the product x ∘_{k} y: = {xa_{k}y}. We have dim(J_{k}) = 3 and {a_{k}, u_{k}, u_{k}^{(2)}} is a basis of J_{k}. Moreover, J_{k} is invariant under the operator g_{k}: = 2(a_{k}□u_{k} − u_{k}□a_{k}) where triple products are computed in Z = L(H). In case s < rank(a) we set J_{n}: = ℝa_{n} as real Jordan algebras.

3.6 Lemma. The Jordan algebras J_{k} and J_{l} with k ≠ l, (k, l = 1, ⋯, r) are orthogonal in the JB^{∗}-triple sense in Z, that is {J_{k}J_{l}Z} = 0.

Proof. For n ∈ {k, l} ⊂ {1, ⋯, s} with k ≠ l, let z_{n} be any element in the basis {a_{n}, u_{n}, u_{n}^{(2)}} of J_{n}. Clearly it suffices to show that z_{k}z_{l} = 0 = z_{l}z_{k}. As an example, we shall prove that u_{k}^{(2)}u_{l}^{(2)} = 0. It is a routine to check that u_{k}u_{l} = 0. Then

u_{k}^{(2)}u_{l}^{(2)} = {u_{k}a_{k}u_{k}} {u_{l}a_{l}u_{l}} = (u_{k}a_{k}u_{k}) (u_{l}a_{l}u_{l}) = u_{k}a_{k}(u_{k}u_{l})a_{l}u_{l} = 0

as we wanted to see. □

Consider now the vector space direct sum J: = ⨁_{1}^{r} J_{k}, and define a product z ∘ w: = {zaw} in J by

z ∘ w: = \{zaw\} = \frac{1}{2}(zaw + waz) = \frac{1}{2}Σ_{1}^{r}(z_{k}a_{k}w_{k} + w_{k}a_{k}z_{k}) = Σ_{1}^{r}z_{k} ∘_{k} w_{k}

where z_{k}, w_{k} are respectively the J_{k}-component of z and w. It is now clear that J is a real Jordan algebra, that the product in J induces in each J_{k} its own product z ∘_{k} w = {za_{k}w} and that the J_{k} are orthogonal as Jordan subalgebras of J. It is also clear that J coincides with the closed real linear span of the set ⋃_{1}^{r}{a_{k}, u_{k}, u_{k}^{(2)}}, in particular dimJ ≤ 3r < ∞. Finally J[a, u] ⊆ J and we conjecture that the equality holds (see [14, prop. 3.5 & th. 3.6].

4 Geodesics and the exponential mapping.

Consider M_{r} endowed with the affine connection ∇ given by (1). To discuss its geodesics, let us define an operator g ∈ Z = L(H) by

g: = g_{a,u}: = 2(u□a − a□u) = 2Σρ_{k}(u_{k}□a_{k} − a_{k}□u_{k}) = Σρ_{k}g_{a_{k},u_{k}}

where u = Σρ_{k}u_{k} is the spectral decomposition of u ∈ Z_{1/2}(a), the a_{k} is any frame associated to the pair (a, u) and g_{k}: = g_{a_{k},u_{k}} is defined in a obvious manner. If the spectral decomposition of u (see (8)) has s < r non zero summands then we define g_{n}: = 0 for n = s + 1, ⋯, r. Then g_{k} is a commutative family of operators in Z, more precisely we have g_{k}(J_{l}) = {0}, g_{k} g_{l} = g_{l} g_{k} = 0 for all k ≠ l, (k, l = 1, ⋯, r) and g leaves invariant all the spaces J and J_{k}. Thus

γ_{a,u}(t): = exp tg(a) = Σ exp tg_{k}(a_{k}), t ∈ ℝ

By section §1 this curve is a geodesic in M_{r} and γ_{a,u}(ℝ) ⊂ J[a, u] ⊂ J. We can collect now the above discussion in the following statement (see [14, prop. 5.1 & 5.4]

4.1 Theorem. Suppose that we are given a point a ∈ M_{r} and a tangent vector u ∈ Z_{1/2}(a)_{s} to M_{r} at a. Then the geodesic of M_{r} that passes through a with velocity u is the curve

γ_{a,u}(t) = Σγ_{a_{k},u_{k}}(t), t ∈ ℝ,

where γ_{k}: = γ_{a_{k},u_{k}} is given by

γ_{k}(t): = γ_{a_{k},u_{k}}(t) = (cos^{2} θ_{k}t) a_{k} + (\frac{1}{2θ_{k}} sin 2θ_{k}t) u_{k} + (\frac{1}{θ_{k}^{2}} sin^{2} θ_{k}t) u_{k}^{(2)}  (G)

Here u = Σρ_{k}u_{k} is the spectral decomposition of u in Z_{1/2}(a), the a_{k} form a frame associated to the pair (a, u) and the numbers θ_{k} are given by cos^{2} θ_{k}: = ρ_{k} with 0 ≤ θ_{k} < \frac{π}{2} .

Now we are in a position to define the exponential mapping. Suppose the tangent vector u lies in the unit ball B_{1}(a) ⊂ Z_{1/2}(a), i.e. ∥u∥ < 1. For t = 1 the expression (G) yields

γ(1) = Σ(cos^{2} θ_{k}) a_{k} + Σ(\frac{1}{2θ_{k}} sin 2θ_{k}) u_{k} + Σ(\frac{1}{θ_{k}^{2}} sin^{2} θ_{k}) u_{k}^{(2)}  (E)

and a real analytic mapping form the unit ball B_{1}(0) ⊂ Z_{1/2}(a) to the manifold M can be defined by

Exp_{a}(u): = γ_{a,u}(1)

An inspection of (E) yields that the Peirce decomposition of γ_{a,u}(1) relative to a is

P_{1}(a)γ_{a,u}(1) = Σ(cos^{2} θ_{k})a_{k},  P_{1/2}(a)γ_{a,u}(1) = Σ(\frac{1}{2θ_{k}} sin^{2} θ_{k}) u_{k}
P_{0}(a)γ_{a,u}(1) = Σ(\frac{1}{θ_{k}^{2}} sin^{2} θ_{k}) u_{k}^{(2)}

Remark that 0 < cos^{2} θ_{k} ≤ 1, hence in particular P_{1}(a)γ_{a,u}(1) lies in the set of all N_{a} of all invertible elements in the JB^{∗}-algebra Z_{1}(a). Clearly N_{a} is an open neighbourhood of a in Z_{1}(a). Remark also that 0 ≤ \frac{1}{2θ_{k}} sin^{2} θ_{k} = ρ_{k} ≤ ∥u∥ < 1, hence Σ(\frac{1}{2θ_{k}} sin^{2} θ_{k}) u_{k} is the spectral decomposition of P_{1/2}(a)γ_{a,u}(1) in Z_{1/2}(a). Thus Exp_{a}B_{1}(a) ⊂ N_{a} ⊂ M. We refer to Exp_{a} as the exponential mapping.

5 Geodesics connecting two given points. The logaritm mapping.

Now we discuss the possibility of joining two given projections a and b such that P_{1}(a)b is invertible in the Jordan algebra Z_{1}(a), by means of a geodesic in M. The remarks in the precedent section show how to proceed. First we compute the spectral decomposition of u: = P_{1/2}(a)b in the JB^{∗}-triple Z_{1/2}(a). Assume it to be

u = P_{1/2}(a)b = Σρ_{k}u_{k},  0 ≤ ρ_{1} ≤ ⋯ ≤ ρ_{r} = ∥u∥ < 1,  1 ≤ k ≤ r

where the u_{k} are pairwise orthogonal minimal tripotents in Z_{1/2}(a). Hence By lemma 3.4 the u_{k} have the form u_{k} = (·, α_{k})ξ_{k}+(·, ξ_{k})α_{k} for some orthonormal families of vectors (α_{k}) ⊂ a(H) and (ξ_{k}) ⊂ a(H)^{⊥}. By lemma 3.2 Q(a_{k})b = {a_{k}ba_{k}} = λ_{k} where λ_{k} ≠ 0 since P_{1}(a)b is invertible in Z_{1}(a). Also |λ_{k}| = ∥\{a_{k}ba_{k}\}∥ ≤ 1. Thus 0 < λ_{k} ≤ 1 and a unique angle 0 ≤ θ_{k} < \frac{π}{2} is determined by cos^{2} θ_{k} = λ_{k}. In this way we have got all the elements appearing in (E). Let us define γ̃(t): = Σγ̃_{k}(t) for t ∈ ℝ where

γ̃_{k}(t): = (cos^{2} tθ_{k}) a_{k} + (\frac{1}{2θ_{k}} sin 2tθ_{k}) u_{k} + (\frac{1}{θ_{k}^{2}} sin^{2} tθ_{k}) u_{k}^{(2)}

By section §1, each γ̃_{k}(t) is a geodesic in the manifold M_{1} of all rank 1 projections. By the previous discussion γ̃_{j}(t) and γ̃_{k}(t) are orthogonal whenever j ≠ k, t ∈ ℝ, hence γ̃(t): = Σγ̃_{k}(t), t ∈ ℝ, is a curve in the manifold M of projections of rank r. Clearly γ̃(0) = Σγ̃_{k}(0) = Σa_{k} = a and we shall now show that b̃: = γ(1) coincides with b. As above P_{1/2}(a) = b̃ = Σ(\frac{1}{2} sin 2θ_{k}) u_{k} = Σρ_{k}u_{k} is the spectral decomposition of P_{1/2}(a)b̃ in Z_{1/2}(a), which by construction is the spectral decomposition of P_{1/2}(a)b. Hence by lemma 3.3, b̃ = γ̃(1) = b. This gives a geodesic γ(t) that connects a with b in the manifold M_{r} and passes through the point a with the velocity u: = P_{1/2}(a)b. It is uniquely determined by the data a, b and the property γ_{a,u}(1) = b.

Now we are in a position to define the logaritm mapping. Fix a point a ∈ M and let N_{a} ⊂ M be the set of all projections b ∈ M such that P_{1}(a)b is invertible in the JB^{∗}-algebra Z_{1}(a). Define a mapping Log_{a} from N_{a} ⊂ M to the unit ball B_{1}(a) ⊂ Z_{1/2}(a) by declaring Log_{a}(b) to be the velocity at t = 0 of the unique geodesic γ_{a,u}(t) that joins a with b in M and γ_{a,u}(1) = b, in other words Log_{a}(b): = P_{1/2}(a)b. We refer to Log_{a} as the logaritm mapping. Clearly Log_{a} and Exp_{a} are real analytic inverse mappings. In particular, the family {(N_{a}, Log_{a}) : a ∈ M} is an atlas of M. We remark the fact that γ_{a,u}[0, 1] ⊂ N_{a} for all u ∈ B_{1}(a) which shall be needed later on to apply the Gauss lemma [11, 1.9] and summarize the above discussion in the statement (see [14, th. 5.7 & prop. 5.8])

5.1 Theorem. Let a and b be two given projections in M_{r} and assume that P_{1}(a)b is invertible in the Jordan algebra Z_{1}(a). Then there is exactly one geodesic γ_{a,u}(t) that joins a with b in M and γ_{a,u}(1) = b.

6 The Riemann structure on M.

Let a ∈ M_{r} and choose any frame (a_{k}) for a. By section §1 we have vector space direct sum decomposition

Z_{1/2}(a) = ⨁_{1}^{r} Z_{1/2}(a_{k})  (9)

which suggests to define a scalar product in Z_{1/2}(a) by

⟨u, v⟩: = \frac{1}{\sqrt{r}} Σ⟨u_{k}, v_{k}⟩_{a_{k}}  (10)

where ⟨·, ·⟩_{a_{k}} stands for the Levi form on Z_{1/2}(a_{k}). First we prove 6.1 Lemma. With the above notation, (9) defines an Aut^{∘}-invariant scalar product on Z_{1/2}(a) that does not depend of the frame a = Σ_{k} and converts Z_{1/2}(a) into a Hilbert space.

Proof. Let Σa_{k} and Σa′_{k} denote two frames for a where a_{k} = (·, α_{k})α_{k} and a′_{k} = (·, α′_{k})α′_{k} for some orthonormal families (α_{k}), (α′_{k}) ⊂ a(H). Extend them to two orthonormal basis of H and let u ∈ L(H) be the unitary operator that exchanges these bases. Then u induces an isometry U ∈ Aut^{∘}(Z) by Uz = uzu^{−1} that satisfies Ua′_{k} = a_{k}. The invariance of the Levi form together with (10) yields part of the result. The remainder is trivial. □

A Riemann structure can now be defined in M_{r} in the following way. Let X, Y ∈ 𝔇(M) vector fields on M_{r}, and for a ∈ M_{r} take any frame a = Σa_{k}. Then (9) gives representation X = ΣX_{k}, Y = ΣY_{k} with X_{k}, Y_{k} ∈ Z_{1/2}(a_{k}) and we set

g_{a}(X, Y): = ⟨X, Y ⟩ = \frac{1}{\sqrt{r}} Σ⟨X_{k}, Y_{k}⟩_{a_{k}} = \frac{1}{\sqrt{r}} Σg_{a_{k}}(X_{k}, Y_{k})

This is a well defined Aut^{∘}-invariant Riemann structure on M_{r}. By section §1 each g_{a_{k}} has property (4) and a routine argument gives the same property for g. Thus g is the only Levi-Civita connection in M_{r} and we can apply the Gauss lemma [11, 1.9] to conclude that the ∇-geodesics are minimizing curves for the Riemann distance.

Recall that for a tripotent a ∈ Z, the mapping σ_{a}: x_{1} + x_{1/2} + x_{0} ↦ x_{1} − x_{1/2} + x_{0}, where x ∈ Z and x_{1} + x_{1/2} + x_{0} is the Peirce decomposition of x with respect to a, called the Peirce symmetry of Z with center a, is an involutory automorphism of Z that induces an isometric symmetry of M_{r} (see [6, th. 5.1]). We let IsomM_{r} and 𝔖 denote the group of all isometries of the Riemann manifold M_{r} and the subgroup generated by the set S: = {σ_{a}: a ∈ M_{r}}, respectively.

6.2 Proposition. With the above notation, M_{r} is a symmetric Riemann manifold in which the group 𝔖 acts transitively.

Proof. Let a, b ∈ M_{r} be such that b ∈ N_{a}. Then a and b can be joined in M_{r} by a unique geodesic with γ(0) = a, γ(1) = b. If c: = γ(\frac{1}{2}), then σ_{c} is a symmetry of M_{r} such that σ_{c}(a) = b. Thus the set S is transitive in N_{a} and S is locally transitive in M_{r}. Consider now the case b ∉ N_{a}. Since M_{r} is pathwise connected, we can join a with b by a curve Γ in M_{r} and by a standard compactness argument there exists a finite set {b_{0}, ⋯, b_{s}} ⊂ Γ[0, 1] such that b_{0} = a, b_{s} = b and b_{k+1} ∈ N_{b_{k}} for k = 1, ⋯, s. An application of the above argument to each pair of consecutive points gives the result. □

We now compute the Riemann distance in M_{r}. Consider first the case of two points a, b ∈ M_{r} with b ∈ N_{a}. Let γ_{a,u}(t) be the unique geodesic that joins a with b in M_{r} and satisfies b = γ_{a,u}(1). Since Aut^{∘}(Z) is transitive in N_{a} and the Levi norm is Aut^{∘}(Z)-invariant, we have

|γ̇_{au}(t)|_{γ_{au}(t)} = |γ̇_{au}(0)|_{γ_{au}(0)} = |u|_{a}

On the other hands, since the Levi norm in Z_{1/2}(a) is the direct hilbertian sum of the Levi norms in the Z_{1/2}(a_{k}), we have by section §1

|u|_{a}^{2} = \frac{1}{r}Σ|u_{k}|_{a_{k}}^{2} = \frac{1}{r}Σθ_{k}^{2}  (D)

where u = Σρ_{k}u_{k} is the spectral decomposition of u in Z_{1/2}(a), (a_{k}) is the frame associated to the pair (a, u) and cos^{2} θ_{k} = ρ_{k}. Therefore

d(a, b) = ∫_{0}^{1} |γ̇_{au}(t)|_{γ_{au}(t)} dt = ∫_{0}^{1} |u|_{a} dt = |u|_{a} = \frac{1}{\sqrt{r}} (Σθ_{k}^{2})^{1/2}

Consider now the case b ∉ N_{a}. By lemma 3.2 we can take a sequence (b_{n})_{n∈ℕ} in N_{a} such that b = lim_{n→∞} b_{n}. since (D) holds for all b_{n} and the Riemann distance is continuous, we get the validity (D) for all a, b ∈ M_{r}. □

Note that expression (D) is a generalization of the classical formula for the Fubini-Study metric in the projective space ℙ(H).

References

1. Chu, C.H., & Isidro, J. M., Manifods of tripotents in JB^{∗}-triples., Math. Z. (to appear).

2. Harris, L. A., Proceedings on Infinite Dimensional Holomorphy., Springer-Verlag, Lecture Notes in Maths. Vol. 364., 1973, p. 13-40.

3. Helgason, S., Differential Geometry and Symmetric Sapces., Academic Press., 1962.

4. Hirzebruch, U., Über Jordan-Algebren und kompakte Riemannsche symmetrische Räume von Rang 1., Math. Z. 90 (1965), 339-354.

5. Isidro, J. M., The manifold of minimal partial isometries in the space L(H, K) of bounded linear operators., Acta Sci. Math. (Szeged) 66 (2000), 451-466.

6. Isidro, J. M., The manifold of minimal tripotents in classical Cartan factors. (to appear).

7. Kaup, W., Über die Automorphismen Grassmanncher Mannigfaltigkeiten unendlicher Dimension., Math. Z. 144 (1975), 75-96.

8. Kaup, W., A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces., Math. Z. 183 (1983), 503-529.

9. Kaup, W., Über die Klassifikation der symmetrischen Hermitesches Mannigfaltigkeiten unendlicher Dimension, I, II., Math. Ann. 257, 262 (1981 and 1983), 463-483 and 503-529.

10. Kaup, W., On Grassmannians associated with JB^{∗}-triples. (to appear).

11. Klingenberg W., Riemannian Geometry., Walter der Gruyter 1982.

12. Loos, O., Bounded symmetric domains and Jordan pairs, Mathematical Lectures, University of California at Irvine 1977.

13. Nomura, T., Manifold of primitive idempotents in a Jordan-Hilbert algebra., J. Math. Soc. Japan 45 (1993), 37-58.

14. Nomura, T., Grassmann manifold of a JH-algebra., Annals of Global Analysis and Geometry 12 (1994), 237-260.

15. Sauter, J., Randstrukturen beschänter symmetrischer Gebiete., Ph. D. Dissertation, Universitát Tübingen 1995.

16. Corach, G., Porta, H., and Recht, L., The Geometry of Spaces of Projections in C^{∗}-algebras., Advances in Math. 101 (1993), 59-77.

17. Kovarik, Z. V., Geodesics and Near-Geodesics in the Manifold of Projector Frames., Linear Algebra Appl. 99 (1988), 259-277.

18. Wilkins, D. R., The Grassmann Manifold of a C^{∗}-algebra., Proc. R. Ir. Acad. 90A (1990), no. 1, 99-116.

Facultad de Matemáticas, Universidad de Santiago, 15706 Santiago de Compostela, SPAIN

E-mail address: jmisidro@zmat.usc.es