Manifolds of algebraic elements in JB^{∗}-triples

José M. Isidro^{∗} Facultad de Matemáticas, Universidad de Santiago, Santiago de Compostela, Spain.

                                jmisidro@zmat.usc.es

October 10, 2001

Abstract

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z = L(H). We represent A as a disjoint union of connected subsets M of Z. Using the algebraic structure of Z, a torsionfree affine connection ∇ (that is invariant under the group Aut(Z) of automorphisms of Z) is defined on each of these connected components and the geodesics are computed. In case M consists of elements that have a fixed finite rank r, (0 < r < ∞), Aut(Z)-invariant Riemann and Kähler structures are defined on M which in this way becomes a totally geodesic symmetric holomorphic manifold. Similar results are established for the manifold of algebraic elements in an abstract JB^{∗}-triple.

Keywords. JB^{∗}-triples, Grassmann manifolds, Riemann manifolds.

AMS 2000 Subject Classification. 48G20, 72H51.

1 Introduction

In this paper we are concerned with the differential geometry of some infinite-dimensional Grassmann manifolds 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 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 [6], 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 [18, 19] established similar results for the manifold of fixed finite rank projections in a topologically simple real Jordan-Hilbert algebra. In [8], the authors studied the Riemann and Kähler structure of the manifold of finite rank projections in Z without the use of any global scalar product. As pointed out there, the Jordan-Banach structure of Z encodes information about the differential geometry of some manifolds naturally associated to it, one of which is the manifold of algebraic elements in Z. On the other hand, the Grassmann manifold of all projections in Z has been discussed by Kaup in [10] and [13]. See also [1, 7] for related results.

It is therefore reasonable to ask whether a Riemann structure can be defined in the set of algebraic elements in Z, and how does it behave when it exists. We restrict our considerations to the set A of all normal algebraic elements in Z that have finite rank. Remark that the assumption concerning the finiteness of the rank can not be dropped, as proved in [8]. Normality allows us to use spectral theory which is an essential tool. In the case H = ℂ^{n}, all elements in Z are algebraic (as any square matrix is a root of its characteristic polynomial) and have finite rank. Under the above restrictions A is represented as a disjoint union of connected subsets M of Z, each of which is invariant under Aut(Z) (the group of all C^{∗}-automorphisms of Z). Using algebraic tools, a holomorphic manifold structure and an Aut(Z)-invariant affine connection ∇ are introduced on M and its geodesics are calculated. One of the novelties is that we take JB^{∗}-triple system approach instead of the Jordan-algebra approach of [18, 19]. As noted in [1] and [7], within this context the algebraic structure of JB^{∗}-triple acts as a substitute for the Jordan algebra structure. In case M consists of elements that have a fixed finite rank r, (0 < r < ∞), the JB^{∗}-triple structure provides a local scalar product known as the algebraic metric of Harris ([2], prop. 9.12). Although Z is not a Hilbert space, the use of the algebraic scalar product allows us to define an Aut(Z)-invariant Riemann and a Kähler structure on M. We prove that ∇ is the Levi-Civita and the Kähler connection of M, and that M is a symmetric holomorphic manifold on which Aut^{∘}(Z) acts transitively as a group of isometries.

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

The role that projections play in the study of the algebra Z = L(H) is taken by tripotents in the study of a JB^{∗}-triple system. A spectral calculus and a notion of algebraic element is available in the stetting of JB^{∗}-triples, and the manifold of all finite rank algebraic elements in a JB^{∗}-triple Z is studied in the final section.

2 Algebraic preliminaries.

For a complex Banach space X denote by X_{ℝ} the underlying real Banach space, and let L(X) and L_{ℝ}(X) respectively be the Banach algebra of all bounded complex-linear operators on X and the Banach algebra of all bounded real-linear operators on X_{ℝ}. 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. For , x, y, z ∈ Z we write L(x, y)(z) = (x□y)(z) and Q(x, y)(z): = {xzy}. Note that L(x, y) ∈ L(Z) whereas Q(x, y) ∈ L_{ℝ}(Z), and that the operators L_{a} = L(a, a) and Q_{a} = Q(a, a) commute. 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 Der(Z) of all 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 and e ∈ Z is called a tripotent if {eee} = e, the set of which is denoted by Tri(Z). For e ∈ Tri(Z), the set of eigenvalues of e□e ∈ L(Z) is contained in \{0, \frac{1}{2}, 1\} and the topological direct sum decomposition, called the Peirce decomposition of Z,

Z = Z_{1}(e) ⊕ Z_{1/2}(e) ⊕ Z_{0}(e).  (1)

holds. Here Z_{k}(e) is the k- eigenspace of e□e 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).

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. In particular, every Peirce space is a JB^{∗}-subtriple of Z and Z_{1}(e)□Z_{0}(e) = {0}. 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). A tripotent e in a JB^{∗}-triple Z is said to be minimal if e ≠ 0 and P_{1}(e)Z = ℂ e, and we let Min(Z) be the set of them. If e ∈ Min(Z) then ∥e∥ = 1. A JB^{∗}-triple Z may have no non-zero tripotents.

Let e = (e_{1}, ⋯, e_{n}) be a finite sequence of non-zero mutually orthogonal tripotents e_{j} ∈ Z, and define for all integers 0 ≤ j, k ≤ n the linear subspaces

\begin{aligned}Z_{j,j}(e) & = Z_{1}(e_{j})  1 ≤ j ≤ n, \\ Z_{j,k}(e) = Z_{k,j}(e) & = Z_{1/2}(e_{j}) ∩ Z_{1/2}(e_{k})  1 ≤ j, k ≤ n, j ≠ k, \\ Z_{0,j}(e) = Z_{j,0}(e) & = Z_{1}(e_{j}) ∩ ⋂_{k≠j} Z_{0}(e_{k})  1 ≤ j ≤ n, \\ Z_{0,0}(e) & = ⋂_{j} Z_{0}(e_{j}). \end{aligned}  (2)

Then the following topologically direct sum decomposition, called the Peirce decomposition relative to e, holds

Z = ⨁_{0≤j≤k≤n} Z_{j,k}(e).  (3)

The Peirce spaces multiply according to the rules {Z_{j,m}Z_{m,n}Z_{n,k}} ⊂ Z_{j,k}, and all products that cannot be brought to this form (after reflecting pairs of indices if necessary) vanish. In terms of this decomposition, the Peirce spaces of the tripotent e: = e_{1} + ⋯ + e_{n} are

\begin{aligned}Z_{1}(e) = ⨁_{j,k} Z_{j,k}(e) & = ( ⨁_{1≤j≤n} Z_{j,j}(e)) ⊕ ( ⨁_{\begin{matrix}1≤j,k≤n \\ j≠k \end{matrix}} Z_{j,k}(e)), \\ Z_{1/2}(e) & = ⨁_{1≤j≤n} Z_{0,j}(e),  Z_{0}(e) = Z_{0,0}(e). \end{aligned}  (4)

Recall that every C*-algebra Z is a JB*-triple with respect to the triple product 2{abc}: = (ab^{∗}c + cb^{∗}a). In that case, every projection in Z is a tripotent and more generally the tripotents are precisely the partial isometries in Z. C^{∗}-algebra derivations and C^{∗}-automorphisms are derivations and automorphisms of Z as a JB^{∗}-triple though the converse is not true.

We refer to [11], [13], [16], [20] and the references therein for the background of JB^{∗}-triples theory.

3 Manifolds of algebraic elements in L(H).

From now on, Z will denote the C^{∗}-algebra L(H). An element a ∈ Z is said to be algebraic if it satisfies the equation p(a) = 0 for some non identically null polynomial p ∈ ℂ[X]. By elementary spectral theory σ(a), the spectrum of a in Z, is a finite set whose elements are roots of the algebraic equation p(λ) = 0. In case a is normal we have

a = ∑_{λ∈σ(a)} λ e_{λ}  (5)

where λ and e_{λ} are, respectively, the spectral values and the corresponding spectral projections of a. If 0 ∈ σ(a) then e_{0}, the projection onto ker(a), satisfies e_{0} ≠ 0 but in (5) the summand 0 e_{0} is null and will be omitted. In particular, in (5) the numbers λ are non-zero pairwise distinct complex numbers and the e_{λ} are pairwise orthogonal non-zero projections. We say that a has finite rank if dim a(H) < ∞, which always occurs if dim(H) < ∞. Set r_{λ}: = rank (e_{λ}). Then a has finite rank if and only if r_{λ} < ∞ for all λ ∈ σ(a)\{0} (the case 0 ∈ σ(a) and dim ker a = ∞ may occur).

Thus, every finite rank normal algebraic element a ∈ Z gives rise to: (i) a positive integer n which is the cardinal of σ(a)\{0}, (ii) an ordered n-uple (λ_{1}, ⋯, λ_{n}) of numbers in ℂ\{0} which is the set of the pairwise distinct non-zero spectral values of a, (iii) an ordered n-uple (e_{1}, ⋯, e_{n}) of non-zero pairwise orthogonal projections, and (iii) an ordered n-uple (r_{1}, ⋯, r_{n}) where r_{k} ∈ ℕ\{0}.

The spectral resolution of a is unique except for the order of the summands in (5), therefore these three n-uples are uniquely determined up to a permutation of the indices (1, ⋯, n). The operator a can be recovered from the set of the first two ordered n-uples, a being given by (5).

Given the n-uples Λ: = (λ_{1}, ⋯, λ_{n}) and R: = (r_{1}, ⋯, r_{n}) in the above conditions, we let

M(n, Λ, R): = { ∑_{k} λ_{k}e_{k} : e_{j}e_{k} = 0 for j ≠ k, rank (e_{k}) = r_{k}, 1 ≤ j, k ≤ n }  (6)

be the set of the elements (5) where the coefficients λ_{k} and ranks r_{k} are given and the e_{k} range over non-zero, pairwise orthogonal projections of rank r_{k}. For instance, for n = 1, Λ = {1} and R = {r} we obtain the manifold of projections with a given finite rank r, that was studied in [8]. For the n-uple Λ = (λ_{1}, ⋯, λ_{n}) we set Λ^{∗}: = (λ̄_{1}, ⋯, λ̄_{n}). The involution z ↦ z^{∗} on Z induces a map M(n, Λ, R) → M(n, Λ^{∗}, R) where M(n, Λ, R)^{∗} = {z^{∗}: z ∈ M} = M(n, Λ^{∗}, R), and Λ ⊂ ℝ if and only if M(n, Λ, R) consists of hermitian elements.

For a normal algebraic element a = ∑_{λ∈σ(a)\{0}} λe_{λ} we define its support to be the projection

a = suppa: = ∑_{λ∈σ(a)\{0}} e_{λ} = e_{1} + ⋯ + e_{n}.

It is clear that h( supp(a)) = supph(a) holds for all h ∈ Aut^{∘}(Z), which combined with the Aut^{∘}(Z)-invariance of Peirce projectors P_{k} gives the following useful formula

P_{k}( supph(a)) = P_{k}(h supp(a)) = h P_{k}( supp(a)) h^{−1},  (k = 1, 1/2, 0).  (7)

Proposition 3.1 Let A be the set of all normal algebraic elements of finite rank in Z, and let M(n, Λ, R) be defined as in (6). Then

A = ⋃_{n, Λ, R} M(n, Λ, R)  (8)

is a disjoint union of Aut^{∘}(Z)-invariant connected subset of Z on which the group Aut^{∘}(Z) acts transitively.

Proof. We have seen before that A ⊂ ⋃_{n, Λ, R} M(n, Λ, R). Conversely, let a belong to some M(n, Λ, R) hence we have a = ∑_{k} λ_{k}e_{k} for some orthogonal projections e_{k}. Then Id = (e_{1} + ⋯ + e_{n}) + f where f is the projection onto ker(a) in case 0 ∈ σ(a) and f = 0 otherwise. The above properties of the e_{k}, f yield easily ap(a) = 0 or p(a) = 0 according to the cases, where p ∈ ℝ[X] is the polynomial p(z) = (z − λ_{1}). · · · .(z − λ_{n}). Hence a ∈ A. Clearly (21) is union of disjoint subsets.

Fix one of the sets M: = M(n, Λ, R) and take any pair a, b ∈ M. Then

a = λ_{1}p_{1} + ⋯ + λ_{n}p_{n}, b = λ_{1}q_{1} + ⋯ + λ_{n}q_{n}.

In case 0 ∈ σ(a), set p_{0} : = Id−∑_{k} p_{k} and q_{0} : = Id−∑_{k} q_{k}. Since rank p_{k} = rank q_{k}, the projections p_{k} and q_{k} are unitarily equivalent and so are p_{0} and q_{0}. Let us choose orthonormal basis B_{k}^{p} and B_{k}^{q} in the ranges p_{k}(H) and q_{k}(H) for k = 0, 1, ⋯, n. Then ⋃_{k} B_{k}^{p} and ⋃_{k} B_{k}^{q} are two orthonormal basis in H. The unitary operator U ∈ Z that exchanges these basis satisfies Ua = b. In particular M is the orbit of any of its points under the action of the unitary group of H. Since this group is connected and its action on Z is continuous, M is connected. □

Let a ∈ Z be a normal algebraic element with finite rank and a = supp(a) its support. In the Peirce decomposition

Z = Z_{1}(a) ⊕ Z_{1/2}(a) ⊕ Z_{0}(a)

every Peirce space Z_{k}(a)_{s} is invariant under the natural involution^{∗} of Z, and we let Z_{k}(a)_{s} denote its selfadjoint part, (k = 1, 1/2, 0). In what follows, the map Z × Z → Z given by (x, y) ↦ g(a, x)y, and the partial maps obtained by fixing one of the variables, will play an important role. For every fixed value x ∈ Z_{1/2}(a), we get an operator g(a, x)(·) which is an inner JB^{∗}-triple derivation of Z, hence we have an operator-valued continuous real-linear map Z_{1/2}(a) → Der(Z). Moreover g(a, x)(·) is a C^{∗}-algebra derivation if and only if x ∈ Z_{1/2}(a)_{s} (see 3.3). For y = a fixed, we get the map x ↦ g(a, x)a for which we introduce the notation

Φ_{a}(x): = g(a, x)a = {a x a} − {x a a} = (Q(a, a) − L(a, a))x, x ∈ Z.

First we discuss Z_{1/2}(a).

Proposition 3.2 Let a ∈ Z be a normal algebraic element of finite rank, and let a = e_{1} + ⋯ + e_{n} be its support. Then Z_{1/2}(a) consists of the operators

u = ∑_{k} u_{k}, u_{k} ∈ Z_{1/2}(e_{k}), e_{k} u_{j} = u_{j} e_{k} = 0, j ≠ k, (1 ≤ j, k ≤ n).  (9)

If u ∈ Z_{1/2}(a)_{s}, then we have the additional condition u_{k} ∈ Z_{1/2}(e_{k})_{s}.

Proof. Let u ∈ Z be selfadjoint. The relation u ∈ Z_{1/2}(a) is equivalent to u = 2{aau} which now reads

u = 2{aau} = aa^{∗}u + ua^{∗}a = ∑_{k}(e_{k}u + ue_{k}) = ∑_{k} u_{k}

where

u_{k}: = e_{k}u + ue_{k} for 1 ≤ k ≤ n.  (10)

Note that e_{j}, e_{k} ∈ Z_{1}(a), hence by the Peirce multiplication rules {e_{j}ue_{k}} ∈ {Z_{1}(a)Z_{1/2}(a)Z_{1}(a)} = {0}, that is e_{j}ue_{k} + e_{k}ue_{j} = 0 for all 1 ≤ j, k ≤ n. Multiplying the latter by e_{j} with j ≠ k yields e_{j}ue_{k} = 0 for j ≠ k, (1 ≤ j, k ≤ n). Therefore by (10),

2{e_{k}e_{k}u_{k}} = e_{k}(e_{k}u + ue_{k}) +(e_{k}u + ue_{k})e_{k} =
(e_{k}u + ue_{k}) + 2e_{k}ue_{k} = (e_{k}u + ue_{k}) = u_{k}

which shows u_{k} ∈ Z_{1/2}(e_{k}) and clearly u_{k} = u_{k}^{∗} for 1 ≤ k ≤ n. Multiplying in (10) by e_{j} with j ≠ k we get u_{k}e_{j} = e_{j}u_{k} = 0 and in particular e_{j}□u_{k} = u_{k}□e_{j} = 0 for j ≠ k.

Conversely, let u_{k} satisfy the properties in (9). Then u: = ∑_{k} u_{k} is selfadjoint and e_{k}u = e_{k} (∑_{j} u_{j}) = e_{k}u_{k}. Similarlyue_{k} = u_{k}e_{k}, hence 2{aau} = aa^{∗}u + ua^{∗}a = (∑_{j} e_{j}) u + u (∑_{j} e_{j}) = ∑_{j}(e_{j}u + ue_{j}) = u,.

Using the ^{∗}-invariance of Z_{1/2}(a) every element in this space can be written in the form u = u_{1} + iu_{2} with u_{1}, u_{2} ∈ Z_{1/2}(a)_{s} and the result follows easily. □

The following result should be compared with ([1], th. 3.1)

Proposition 3.3 Let a ∈ Z be a normal algebraic element of finite rank and a: = supp(a). Then for any u ∈ Z_{1/2}(a), the operator g(a, u): = a□u − u□a is an inner C^{∗}-derivation of Z if and only if u is selfadjoint.

Proof. Let a = ∑_{k} λ_{k}e_{k} and a = ∑_{k} e_{k} be the spectral resolution and the support of a. Suppose u = u^{∗}. By (3.2) u has the form u = ∑ u_{k} with u_{k} ∈ Z_{1/2}(e_{k})_{s} and e_{k}□u_{j} = u_{j}□e_{k} = 0 for all j ≠ k. Therefore

g(a, u) = ∑_{k}(e_{k}□u_{k} − u_{k}□e_{k}) = ∑_{k} g(e_{k}, u_{k}).  (11)

Here the e_{k} are projections in Z and u_{k} ∈ Z_{1/2}(e_{k})_{s}, hence by ([1], th. 3.1) each g(e_{k}, u_{k}) is an inner C^{∗}- derivation of Z and so is the sum. Conversely, since a is a projection, whenever g(a, u) is a C^{∗}-algebra derivation we have u ∈ Z_{1/2}(a)_{s} again by ([1], th. 3.1). □

Now consider the joint Peirce decomposition of Z relative to the family (e_{1}, ⋯, e_{n}) where a = λ_{1}e_{1} + ⋯ + λ_{n}e_{n} is the spectral resolution of a. Remark that ⨁_{1≤k≤n} i A(e_{k}) ⊂ Z_{1}(a) is a direct summand of Z, hence so is the space

X: = ( ⨁_{1≤k≤n} i A(e_{k})) ⊕ Z_{1/2}(a).

Proposition 3.4 Let a ∈ Z be a normal algebraic element of finite rank and a: = supp(a). Then Φ_{a} is a surjective complex linear homeomorphism of Z_{1/2}(a). If a is hermitian then Φ_{a} is a surjective real linear homeomorphism of X that preserves the subspace ⨁_{1≤k≤n} i A(e_{k}).

Proof. Let x = iv + u ∈ X where v ∈ ⨁_{1≤k≤n} A(e_{k}) and u ∈ Z_{1/2}(a). The Peirce multiplication rules give for v = ∑_{j} v_{j} with v_{j} ∈ A(e_{j}) and u = ∑_{k} u_{k} according to (3.2)

{aZ_{1/2}(a)a} = {0},

{a iv a} = −i{∑_{j} e_{j} ∑_{k} v_{k} ∑_{l} λ_{l}e_{l}} = −i ∑_{k} λ_{k}v_{k},

\{u a a\} = i\{∑_{j} u_{j} ∑_{k} e_{k} ∑_{l} λ_{l}e_{l}\} = \frac{i}{2} ∑_{k} λ_{k}u_{k}.

Therefore

Φ_{a}(x) = −2i ∑_{k} λ_{k}v_{k} − \frac{1}{2} ∑_{k} λ_{k}u_{k} ∈ ( ⨁_{1≤k≤n} Z_{1}(e_{k})) ⊕ Z_{1/2}(a).  (12)

It is now clear that Φ_{a} preserves Z_{1/2}(a). If a is hermitian then Λ ⊂ ℝ^{n} and Φ_{a} also preserves ⨁_{1≤k≤n} i A(e_{k}). Moreover Φ_{a}(x) = 0 with x ∈ X is equivalent to ∑ λ_{k}v_{k} = 0 = ∑ λ_{k}u_{k} which is equivalent to v = 0 = u since the coefficients satisfy λ_{k} ∈ σ(a)\{0}. We can recover x from Φ_{a}(x), hence the result follows. □

Recall that a subset M ⊂ Z is called a real analytic (respectively, holomorphic) submanifold if to every a ∈ M there are open subsets P, Q ⊂ Z and a closed real-linear (resp. complex) subspace X ⊂ Z with a ∈ P and ϕ(P ∩ M) = Q ∩ X for some bianalytic (resp. biholomorphic) map ϕ: P → Q. If to every a ∈ M the linear subspace X = T_{a}M, called the tangent space to M at a, can be chosen to be topologically complemented in Z then M is called a direct submanifold of Z.

Fix one of the sets M = M(n, Λ, R) and a point a ∈ M with spectral resolution a = ∑_{k} λ_{k}e_{k}. By the orthogonality properties of the e_{k}, the successive powers of a have the expression

a^{l} = λ_{1}^{l}e_{1} + ⋯ + λ_{n}^{l}e_{n}, 1 ≤ l ≤ n,

where the determinant det(λ_{k}^{l}) ≠ 0 does not vanish since it is a Vandermonde determinant and the λ_{k} are pairwise distinct. Thus the e_{k} are polynomials in a whose coefficients are rational functions of the λ_{k}. Suppose M is a differentiable manifold, and let us obtain its tangent space T_{a}M. Consider a smooth curve t ↦ a(t) through a ∈ M, t ∈ I, for a neighbourhood I of 0 ∈ ℝ and a(0) = a. Each a(t) has a spectral resolution

a(t) = λ_{1}e_{1}(t) + ⋯ + λ_{n}e_{n}(t),

therefore the maps t ↦ e_{k}(t), (1 ≤ k ≤ n), are smooth curves in the manifolds 𝔐(r_{k}) of the projections in Z that have fixed finite rank r_{k} = rank (e_{k}), whose tangent spaces at e_{k} = e_{k}(0) are Z_{1/2}(e_{k}) (see [1] or [8]). Therefore

u_{k}: = \frac{d}{dt}|_{t=0}e_{k}(t) ∈ Z_{1/2}(e_{k}), 1 ≤ k ≤ n.

Since the spectral projections of a(t) corresponding to different spectral values λ_{k} ≠ λ_{j} are orthogonal, we have e_{j}(t) e_{k}(t) = 0 for all t ∈ I, and taking the derivative at t = 0,

e_{j} u_{k} = u_{k} ej = 0,  j ≠ k, 1 ≤ j, k ≤ n.  (13)

By 19, the tangent vector to t ↦ a(t) at t = 0, that is, u: = \frac{d}{dt}|_{t=0}a(t) = ∑_{k} λ_{k}u_{k} satisfies

{a a u} = {∑_{j} e_{j} ∑_{k} e_{k} ∑_{l} λ_{l}u_{l}} = ∑_{j,k,l} λ_{l} {e_{j}e_{k}u_{l}} =

∑_{k,l} λ_{l}\{e_{k}e_{k}u_{l}\} = \frac{1}{2} ∑_{k,l} λ_{l}(e_{k}u_{l} + u_{l}e_{k}) = ∑_{l} λ_{l}\{e_{l}e_{l}u_{l}\} = \frac{1}{2} ∑_{l} λ_{l}u_{l} = \frac{1}{2}u

hence u ∈ Z_{1/2}(a), and T_{a}M can be identified with a vector subspace of Z_{1/2}(a). In fact T_{a}M = Z_{1/2}(a) as it easily follows from the following result that should be compared with ([1] th. 3.3)

Theorem 3.5 The sets M = M(n, Λ, R) defined in (6) are holomorphic direct submanifolds of Z. The tangent space at the point a ∈ M is the Peirce subspace Z_{1/2}(a) where a = supp(a), and a local chart at a given by

f: u ↦ f(u): = (exp g(a, u))a  (14)

with g(a, u) = a□u − u□a.

Proof. M ⊂ Z is invariant under Aut^{∘}(Z). Fix any a ∈ M and let X: = ( ⨁_{1≤k≤n} i A(e_{k})) ⊕ Z_{1/2}(a). Thus Z = X ⊕ Y for a certain subspace Y. The mapping X ⊕ Y → Z defined by (x, y) ↦ F (x, y): = (exp g(a, x))y ∈ Z is a real-analytic and its Fréchet derivative at (0, a) is invertible. In fact this derivative is

\frac{∂F}{∂x} |_{(0,a)}(u, v) = g(a, u)a = Φ_{a}(u),
\frac{∂F}{∂y} |_{(0,a)}(u, v) = ( exp g(a, 0))v = v,

which is invertible according to (3.4). By the implicit function theorem there are open sets U, V with 0 ∈ U ⊂ X and a ∈ V ⊂ Y such that W: = F (U × V) is open in Z and F: U × V → W is bianalytic.

To simplify notation set X_{1} = Z_{1/2}(a) ⊂ X. Then f = F |X_{1} establishes a real analytic homeomorphism between the sets N_{1} : = U ∩ X_{1} and M_{1} : = f(N_{1}). Since X_{1} is a direct summand in X (hence also in Z), the image M_{1} = f(N_{1}) is a direct submanifold.

The operator g(a, x) = a□x − x□a is an inner JB^{∗}triple derivation of Z, hence h: = exp g(a, u) is a JB^{∗}-triple automorphism of Z. Actually h lies in Aut^{∘}(Z), the identity connected component. But it is known ([10]) that Aut(Z) has two connected components and that the elements in the identity component are C^{∗}-algebra automorphisms of Z since they have the form z ↦ UzU^{∗} for some U in the unitary group of H. In particular h preserves normality, spectral values and ranks hence it preserves M and so

M_{1} = f(N_{1}) = {(exp g(a, u))a : u ∈ N_{1}} ⊂ M.

To complete the proof, it suffices to show that f = F |X_{1} is a biholomorphic mapping. The Fréchet derivative of f at a is

f′|_{a}(u) = g(a, u)a = {a, u, a} − {u, a, a}, u ∈ Z_{1/2}(a).

Therefore \overline{∂}f′u = \{a, u, a\} and ∂f′u = −\{u, a, a\} are the (uniquely determined) complex-linear and complexantilinear components of f′u. The Peirce rules give {a, u a} = 0 for all u ∈ Z_{1/2}(a), hence f is holomorphic and the same argument holds for the inverse f^{1} map. □

Remark that if the algebraic element a is a projection then a = a and M as a differentiable manifold is the one constructed in ([1] th. 3.3) and [8].

4 The Jordan connection on M(n, Λ, R)

Let a ∈ M: = M(n, Λ, R) and set a = supp(a). Recall that a vector field X on M is a map from M to the tangent bundle T M. Thus X_{a}, the value of X at a ∈ M, satisfies X_{a} ∈ T_{a}M ≈ Z_{1/2}(a). We let 𝔇(M) be the Lie algebra of smooth vector fields on M. Since the tangent space T_{a}M at a ∈ M has been identified with Z_{1/2}(a), we shall consider every vector field on M as a Z-valued function such that the value at a is contained in Z_{1/2}(a). Let Y′_{a} be the Fréchet derivative of Y ∈ 𝔇(M) at a. Thus Y′_{a} is a bounded linear operator Z_{1/2}(a) → Z, hence Y′_{a}X_{a} ∈ Z and it makes sense to take the projection P_{1/2}(a)Y′_{a}X_{a} ∈ Z_{1/2}(a) ≈ T_{a}M.

Definition 4.1 We define a connection ∇ on M by

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

Note that if a is a projection, then a = supp(a) and ∇ coincides with the affine connection defined in ([1] def 3.6) and [8]. It is a matter of routine to check that ∇ is an affine connection on M, that it is Aut^{∘}(Z)- invariant and torsion-free, i. e.,

g (∇_{X}Y) = ∇_{g(X)} g (Y), g ∈ Aut^{∘}(Z),

where (g X)_{a}: = g′_{a} (X_{g_{a}^{−1}}) for all X ∈ 𝔇(M), and

T (X, Y): = ∇_{X}Y − ∇_{Y} X − [XY] = 0, X, Y ∈ 𝔇(M).

Theorem 4.2 Let the manifolds M be defined as in (6). Then the ∇-geodesics of M are the curves

γ(t): = (exp t g(a, u))a,  t ∈ ℝ,  (15)

where a ∈ M and u ∈ Z_{1/2}(a).

Proof. Recall that the geodesics of ∇ are the curves t ↦ γ(t) ∈ M that satisfy the second order ordinary differential equation

(∇_{γ̇(t)} γ̇(t))_{γ(t)} = 0.

Let u ∈ Z_{1/2}(a). Then g(a, u) = a□u − u□a is an inner JB^{∗}-triple derivation of Z, and, as established in the proof of (3.5), h(t): = exp t g(a, u) is an inner C^{∗}-automorphism of Z. Thus h(t)a ∈ M and t ↦ γ(t) is a curve in the manifold M. Clearly γ(0) = a and taking the derivative with respect to t at t = 0 we get by the Peirce rules

γ̇(t) = g(a, u)γ(t) = h(t)g(a, u)a,  γ̇(0) = g(a, u)a ∈ Z_{1/2}(a),
γ̈(t) = g^{2}(a, u)γ(t) = h(t)g(a, u)^{2}a,  γ̈(0) = g(a, u)γ̇(0) ∈ Z_{1}(a) ⊕ Z_{0}(a).

In particular P_{1/2}(a)g(a, u)^{2}a = 0. The definition of ∇ and the relation (7) give

(∇_{γ̇(t)} γ̇(t))_{γ(t)}  = P_{1/2}( suppγ(t)) (γ̇(t)′_{γ(t)} γ̇(t)) = P_{1/2}( suppγ(t)) γ̈(t) =
P_{1/2}( supph(t)a) h(t)g(a, u)a = h(t)P_{1/2}( supp(a)) g(a, u)^{2}a = 0

for all t ∈ ℝ. Using the representation u = ∑_{k} u_{k} given by (9) one gets g(a, u)a = −\frac{1}{2} ∑_{k} λ_{k}u_{k}, and as λ ∈ σ(a)\{0} the mapping u ↦ g(a, u)a is a linear homeomorphism of Z_{1/2}(a). Since geodesics are uniquely determined by the initial point γ(0) = a and the initial velocity γ̇(0) = g(a, u)a, the above shows that family of curves in (15) with a ∈ M and u ∈ T_{a}M ≈ Z_{1/2}(a) are all geodesics of the connection ∇. □

Recall that a = supp(a) is a finite rank projection, hence by ([8], th. 1.1) the JB^{∗}-subtriple Z_{1/2}(a) has finite rank and the tangent space T_{a}M ≈ Z_{1/2}(a) is linearly homeomorphic to a Hilbert space under an Aut^{∘}(Z)- invariant scalar product (say ⟨· , ·⟩). Thus we can define a Riemann metric on M by

g_{a}(X, Y): = ⟨X_{a}, Y_{a}⟩,  X, Y ∈ 𝔇(M),  a ∈ M.  (16)

Remark that g is hermitian, i.e. we have g_{a}(iX, iY) = g_{a}(X, Y), and that it has been defined in algebraic terms, hence it is Aut^{∘}(Z)-invariant. Moreover, ∇ is compatible with the Riemann structure, i. e.

X g(Y, W) = g(∇_{X} Y, W) + g(Y, ∇_{X} W), X, Y, W ∈ 𝔇(M).

Therefore, ∇ is the only Levi-Civita connection on M. On the other hand, let the map J: Z_{1/2}(a) → Z_{1/2}(a) be given by Jz: = iz. Clearly J^{2} = −Id, hence J defines (the usual) complex structure on the tangent space to M and ∇ is J-hermitian

∇_{X} (iY) = i∇_{X} Y, X, Y ∈ 𝔇(M),

hence ∇ is the only hermitian connection on M. Thus the Levi-Civita and the hermitian connection are the same in this case, and so ∇ is the Kähler connection on M.

For a tripotent e ∈ Tri(Z), the Peirce reflection around e is the linear map S_{e}: = Id − P_{1/2}(e) or in detail z = z_{1} + z_{1/2} + z_{0} ↦ S_{e}(z) = z_{1} − z_{1/2} + z_{0} where z_{k} are the Peirce e-projections of z, (k = 1, 1/2, 0). Recall that S_{e} is an involutory triple automorphism of Z with S_{e}(e) = e, and that if e is a projection (taken as a tripotent) then S_{e} is a C^{∗}-algebra automorphism of Z. This applies to a = supp(a), hence to each a ∈ M we get S_{a}, an involutory automorphism of the manifold M which in this way becomes a symmetric holomorphic Riemann (Kähler) manifold. Note that in general a ∉ M even if a ∈ M, hence S_{a} may have no fixed points in M.

It would be interesting to know if any two points a, b in M can be joined by a geodesic and whether geodesics are minimizing curves for the Riemann distance. The answers to these questions are affirmative when M consists of projections of the same finite rank (see [8]).

5 Algebraic elements in JB^{∗}-triples

The role that projections play in the study of algebras is taken by tripotents in the study of triple systems. A spectral calculus and a notion of algebraic elements is available in the stetting of JB^{∗}-triples. In what follows we shall consider the manifold of all finite rank algebraic elements in a JB^{∗}-triple Z.

Definition 5.1 An element a ∈ Z is called algebraic if there exits a decomposition

a = λ_{1}e_{1} + ⋯ + λ_{n}e_{n}  (17)

where (e_{k}) is a family of pairwise orthogonal tripotents in Z and (λ_{k}) are complex coefficients.

For an algebraic element a ∈ Z the above decomposition can always be chosen in such a way that every e_{k} is non-zero and the λ_{k} are real numbers with 0 < λ_{1} < ⋯ λ_{n}, and under these additional conditions the spectral representation of a is unique. Clearly a has finite rank if and only if every so does every e_{k}.

Remark that for Z = L(H), normal algebraic elements in the C^{∗}-algebra Z are algebraic elements in Z as a JB^{∗}-triple. Given a positive integer n ∈ ℕ, an increasing n-uple of non-zero real numbers Λ = (λ_{1}, ⋯, λ_{n}) and an n-uple R = (r_{1}, ⋯, r_{n}) where 0 < r_{k} ∈ ℕ, we define

N(n, Λ, R): = { ∑_{k} λ_{k}e_{k} : e_{j}□e_{k} = 0 for j ≠ k, rank (e_{k}) = r_{k}, 1 ≤ j, k ≤ n }  (18)

to be the set of the elements (17) where the coefficients λ_{k} and ranks r_{k} are given and the e_{k} range over non-zero, pairwise orthogonal tripotents in Z such that rank (e_{k}) = r_{k}. The set A of finite rank algebraic elements in Z is the disjoint union A = ∪_{n, Λ, R}N(n, Λ, R).

Lemma 5.2 Let Z be an irreducible JBW^{∗}-triple. Then each of sets N = N(, n Λ, R) is an Aut^{∘}(Z)-invariant connected subset of Z on which the group Aut^{∘}(Z) acts transitively.

Proof. Irreducible JBW^{∗}-triples are Cartan factors and we may assume that Z is a not special as otherwise dim Z < ∞ and the result is known [16]. Thus Z is a J^{∗}-algebra in the sense of Harris [4] that is, a weak*-operator closed complex linear subspace of L(H, K) that is closed under the operation of taking triple products, for suitable complex Hilbert spaces H, K with dim H ≤ dim K. Tripotents are the partial isometries e: H → K that lie in Z.

We make a type by type proof. Let Z = L(H, K) be a type I Cartan factor and let a, b ∈ N. In particular

a = λ_{1}e_{1} + ⋯ + λ_{n}e_{n}, b = λ_{1}e′_{1} + ⋯ + λ_{n}e′_{n}

Let H_{k}, H′_{k} ⊂ H be the domains of the partial isometries e_{k} and e′_{k}, and similarly let K_{k}, K′_{k} ⊂ K denote their respective ranges. Since e_{k} and e′_{k} have the same finite rank r_{k}, they are unitarily equivalent, that is there are unitary operators U_{k}: H_{k} → H′_{k} and V_{k}: K_{k} → K_{k} such that e′_{k} = V_{k}e_{k}U_{k}. Since the e_{k} are pairwise orthogonal we have H_{k} ⊥ H_{j} and K_{k} ⊥ K_{j} for k ≠ j and ⨁ U_{k}, ⨁ V_{k} are unitary operators on ⨁ H_{k} and ⨁ K_{k} that can be extended to unitary operators U: H → H and V : K → K if needed. The mapping Z → Z given by z ↦ V zU is a JB^{∗}-triple automorphism that lies in Aut^{∘}(Z) [10] and clearly satisfies b = V aU. Hence Aut^{∘}(Z) acts transitively on N, N is connected and invariant under that group.

Cartan factors of types II and III can treated in the same way. The case of spin factors may be discussed with a different approach, but we shall not go into details. □

Now consider the joint Peirce decomposition of Z relative to the family (e_{1}, ⋯, e_{n}) where a = λ_{1}e_{1} + ⋯ + λ_{n}e_{n} is the spectral resolution of a. Let the support of a be tripotent a = suppa: = e_{1} + ⋯ + e_{n}, and note that

X: = ( ⨁_{1≤k≤n} i A(e_{k})) ⊕ Z_{1/2}(a).

is a topologically complemented subspace in Z.

Fix one of the sets N = N(n, Λ, R) and a point a ∈ N with spectral resolution a = ∑_{k} λ_{k}e_{k}. From the properties e_{k}□e_{j} = 0 for j ≠ k, the successive odd powers of a have the expression

a^{l} = λ_{1}^{2l+1}e_{1} + ⋯ + λ_{n}^{2l+1}e_{n}, 0 ≤ l ≤ n − 1,

where the determinant det(λ_{k}^{2l+1}) ≠ 0 does not vanish since it is a Vandermonde determinant and the λ_{k} are pairwise distinct. Thus the e_{k} are polynomials in a whose coefficients are rational functions of the λ_{k}. Suppose N is a differentiable manifold, and let us obtain its tangent space T_{a}N. Consider a smooth curve t ↦ a(t) in N through a, t ∈ I, for a neighbourhood I of 0 ∈ ℝ and a(0) = a. Each a(t) has a spectral resolution

a(t) = λ_{1}e_{1}(t) + ⋯ + λ_{n}e_{n}(t),

therefore the maps t ↦ e_{k}(t), (1 ≤ k ≤ n), are smooth curves in the manifolds 𝔑(r_{k}) of the tripotents in Z that have fixed finite rank r_{k} = rank (e_{k}), whose tangent spaces at e_{k} = e_{k}(0) are respectively i A(e_{k}) ⊕ Z_{1/2}(e_{k}) (see [1] or [8]). Therefore

z_{k}: = \frac{d}{dt}|_{t=0} e_{k}(t) = iv_{k} + u_{k}: ∈ i A(e_{k}) ⊕ Z_{1/2}(e_{k}), 1 ≤ k ≤ n.

Set v: = ∑_{k} λ_{k}v_{k} and u: = ∑_{k} λ_{k}u_{k}. From Z_{1}(e_{k})□Z_{0}(e_{j}) = {0}, we get

{a a iv} = i ∑_{j,k,l} λ_{l}{e_{j}e_{k}v_{l}} = i ∑_{k} λ_{k}v_{k} = iv ∈ i ⨁_{k} A(e_{k})

The spectral tripotents of a(t) corresponding to different spectral values λ_{k} ≠ λ_{j} are orthogonal, hence e_{j}(t)□e_{k}(t) = 0 for all t ∈ I, and taking the derivative at t = 0 we get

e_{j}□u_{k} = u_{k}□ej = 0,  j ≠ k, 1 ≤ j, k ≤ n.  (19)

Hence

\{a a u\} = \{∑_{j} e_{j} ∑_{k} e_{k} ∑_{l} λ_{l}u_{l}\} = ∑_{j,k,l} λ_{l}\{e_{j}e_{k}u_{l}\} = \frac{1}{2} ∑_{k} λ_{k}u_{k} = \frac{1}{2}u

which shows that u ∈ Z_{1/2}(a). By 19, the tangent vector to t ↦ a(t) at t = 0 is z: = \frac{d}{dt}|_{t=0}a(t) = ∑_{k} λ_{k}(iv_{k} + u_{k}) = iv + u hence it satisfies

\{a a z\} = iv + \frac{1}{2}u ∈ i ⨁ A(e_{k}) ⊕ Z_{1/2}(a),

hence T_{a}N can be identified with a vector subspace of i ⨁ A(e_{k}) ⊕ Z_{1/2}(a). In fact T_{a}N coincides with that space as it easily follows from the following result that should be compared with ([1] th. 3.3)

Theorem 5.3 The sets N = N(n, Λ, R) defined in (18) are real analytic direct submanifolds of Z. The tangent space at the point a ∈ N is the Peirce subspace X, where a = supp(a), and a local chart at a given by

f: z ↦ f(z): = (exp g(a, z))a  (20)

with g(a, z) = a□z − z□a.

Proof. N ⊂ Z is invariant under Aut^{∘}(Z). Fix any a ∈ N and let X: = ( ⨁_{1≤k≤n} i A(e_{k}))⊕Z_{1/2}(a). Thus Z = X⊕Y for a certain subspace Y. The mapping X ⊕ Y → Z defined by (x, y) ↦ F (x, y): = (exp g(a, x))y ∈ Z is a real-analytic and its Fréchet derivative at (0, a) is invertible as proved in (3.4). By the implicit function theorem there are open sets U, V with 0 ∈ U ⊂ X and a ∈ V ⊂ Y such that W: = F (U × V) is open in Z and F: U × V → W is bianalytic and the image F (U) is a direct real analytic submanifold of Z.

The operator g(a, z) = a□z − z□a is an inner JB^{∗}triple derivation of Z, hence h: = exp g(a, z) is a JB^{∗}- triple automorphism of Z. Actually h lies in Aut^{∘}(Z), the identity connected component. In particular h preserves the algebraic character and the spectral decomposition, hence it preserves N and so

F (N) = {(exp g(a, z))a : z ∈ U} ⊂ N.

This completes the proof. □

Definition 5.4 For the tripotents e, e′ we set e ∼ e′ if and only if e and e′ have the same k-Peirce projectors for k = 0, 1/2, 1.

This notion was introduced by Neher who proved ([17], th.2.3) that

e ∼ e′ ⟺ e ∈ Z_{1}(e′) and e′ ∈ Z_{1}(e),  (21)

or equivalently if and only if e□e = e′□e′. Next we extend this relation to an equivalence in the manifold N.

Definition 5.5 Let a, b be elements in N with spectral resolutions a = ∑_{k} λ_{k}e_{k} and b = ∑_{k} λ_{k}f_{k} respectively. We say that a and b are equivalent (and write a ∼ b) if the joint Peirce decompositions of Z relative to the orthogonal families E = (e_{k}) and F = (f_{k}) are the same.

Note that ∼ coincides with the equivalence of Neher when the algebraic elements a and b are tripotents. By ([16], th. 3.14), the Peirce spaces of the tripotent e_{k} can be expressed in terms of the joint Peirce decomposition of Z relative to E, hence a ∼ b if and only if e_{k} ∼ f_{k} for 1 ≤ k ≤ n.

Proposition 5.6 Let a, b be points in N such that a = ∑ λ_{k}e_{k} and b = (exp g(a, z)a for some tangent vector z = iv + u ∈ ( ⨁_{1≤k≤n} i A(e_{k})) ⊕ Z_{1/2}(a). Then a ∼ b if and only if u = 0.

Proof. Let b = (exp g(a, z)a = ∑_{k} λ_{k}f_{k} be the spectral resolution of b. Then each f_{k} is an odd polynomial in b, say f_{k} = p_{k}(b), 1 ≤ k ≤ n. To simplify the notation, consider the index k = 1 and omit the reference to it in the rest of the proof. If a ∼ b then e ∼ f hence by (21) we must have f = {eef} that is

p(b) = {eep(b)} = p({eeb})  (22)

Clearly we have ρb ∼ a for all ρ ∈ 𝕋, which replaced above yields an identity between two polynomials in ρ. Let X^{m}, for some positive odd integer m, be the term of p of lowest degree whose coefficient is not zero. Then (22) entails b^{m} = {eeb^{m}}, that is (exp g(a, z))^{m}a = {e e (exp g(a, z))^{m}a}. Taking the Fréchet derivative at the origin g(a, ·) a = {e e g(a, ·) a}, which evaluated at the tangent vector z = iv + u = i ∑_{k} v_{k} + ∑_{k} u_{k} and using the Peirce rules as in the proof of (3.4) yields u = 0. The converse is easy. □

In particular, there is a neighbourhood of a in N in which the algebraic elements b equivalent to a are those of the form b = (exp g(a, iv)) a with v = ∑_{k} v_{k} ∈ ⨁_{k} A(e_{k}), which gives the expression of the fibre of N through a.

Proposition 5.7 Let a ∈ N be an algebraic element in Z with spectral resolution a = ∑_{k} λ_{k}e_{k}. Then the fibre of N through a is the set of the elements ∑_{k} λ_{k}z_{k} where z_{k} lies in the unit circle of the JB^{∗}-algebra Z_{1}(e_{k}) for 1 ≤ k ≤ n.

Proof. Let v = ∑_{k} λ_{k}v_{k} ∈ ⨁_{k} A(e_{k}), and consider the curves in Z

ϕ(t): = (exp tg(a, iv))a,  ψ(t): = ∑_{k} λ_{k}(exp tg(e_{k}, iv_{k}))e_{k}: = ∑_{k} λ_{k}ψ_{k}(t),  t ∈ ℝ.

They are the solutions of the differential equations

\frac{dϕ(t)}{dt} = g(a, ϕ(t)), \frac{dψ(t)}{dt} = ∑_{k} λ_{k}g(e_{k}, ψ_{k}(t))

with the initial conditions ϕ(0) = a and ψ(0) = ∑_{k} λ_{k}e_{k} = a respectively. From Z_{1}(e_{k})□Z_{1}(e_{j}) = {0} for k ≠ j we get

g(a, iv) = g(∑_{k} e_{k}, i ∑_{j} λ_{j}v_{j}) = ∑_{k} λ_{k}g(e_{k}, iv_{k})

and the uniqueness of solutions of differential equations gives ϕ(t) = ∑_{k} λ_{k}ψ_{k}(t) for all t ∈ ℝ. But it is known ([16] th. 5.6) that for fixed k, 1 ≤ k ≤ n, the set z_{k} = (exp tg(e_{k}, iv_{k}))e_{k}, t ∈ ℝ, v_{k} ∈ A(e_{k}), is the unit circle of the JB^{∗}-algebra Z_{1}(e_{k}), that is the set of those w ∈ Z_{1}(e_{k}) that satisfy w^{∗} = w^{−1}. This completes the proof. □

By restricting the local charts in (20) to the direct summand Z_{1/2}(a) ⊂ T_{a}N we get a direct submanifold B = B(n, Λ, R) of Z, and we refer to B as the base manifold of N. Clearly B is a holomorphic submanifold of the real analytic manifold N, and as in section 3

(∇_{X} Y )_{a}: = P_{1/2}(a)Y′_{a}X_{a}, X, Y ∈ 𝔇(B), a ∈ B,

is an Aut^{∘}(Z)-invariant torsionfree affine connection on B whose geodesics are the curves γ(t): = (exp t g(a, u))a, t ∈ ℝ, for a ∈ B and u ∈ Z_{1/2}(a). Moreover, for a ∈ B the Peirce reflection with respect to a is an involutory triple automorphisms of Z that fixes a, hence it fixes i ⨁_{k} A(e_{k}) and Z_{1/2}(a). It is easy to see that this reflection commutes with the exponential mapping, hence it fixes B(n, Λ, R) and os it defines a holomorphic symmetry of B. In general (a) does not belong to B hence this symmetry in general has no fixed points in B. When the algebraic element a ∈ Z has finite rank, that is when rank (a) = ∑_{k} rank (e_{k}) < ∞, the subtriple Z_{1/2}(a) is linearly equivalent to a complex Hilbert space by [12] and by using the algebraic metric of Harris one can introduce an Aut^{∘}(Z)-invariant Riemann structure and a Kähler structure on the base manifold in exactly the same way we did in section 3, and the connection ∇ turns out to be the Levi-Civita and the Kähler connection on B.

References

[1] Chu, C.H., & Isidro, J. M. Manifolds of tripotents in JB^{∗}-triples Math. Z. 233 2000, 741-754.

[2] Dineen, S. The Schwarz lemma, Oxford Mathematical Monographs, Clarendon Press, Oxford 1989.

[3] Hansche-Olsen, H & Stormer, E. Jordan Operator Algebras, Monographs and Studies in Mathematics vol 21, Pitmann, Boston 1984.

[4] Harris, L. A. Bounded symmetric homogeneous domains in infinite dimensional spaces. In: Proceedings on Infinite dimensional Holomorphy, Lecture Notes in Mathematics 364 1973, 13-40, Springer-Verlag Berlin 1973.

[5] Harris, L. A. & Kaup, W. Linear algebraic groups in infinite dimensions. Illinois J. Math. 21 1977, 666-674.

[6] Hirzebruch, U. Über Jordan-Algebren und kompakte Riemannsche symmetrische Räume von Rang 1. Math. Z. 90 1965, 339-354.

[7] 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, 793-808.

[8] Isidro, J. M. & Mackey, M. The manifold of finite rank projections in the algebra L(H) of bounded linear operators. To appear in Exp. Math.

[9] Isidro, J. M. & Stachó, L. L. The manifold of finite rank tripotents in JB^{∗}-triples To appear.

[10] Kaup, W. Über die Automorphismen Grassmannscher Mannigfaltigkeiten unendlicher Dimension. Math. Z. 144 1975, 75-96.

[11] Kaup, W. A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183 1983, 503-529.

[12] Kaup, W. Über die Klassifikation der symmetrischen Hermiteschen Mannigfaltigkeiten unendlicher Dimension, I, II. Math. Ann. 257 1981, 463-483 and 262 1983, 503-529.

[13] Kaup, W. On Grassmannians associated with JB^{∗}-triples. Math. Z. 236 2001, 567-584.

[14] Kaup, W. Cauchy-Riemann structures associated with bounded symmetric domains. preprint 2001,

[15] Klingenberg W. Riemannian Geometry, Walter der Gruyter 1982.

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

[17] Neher, E. Grids in Jordan triple systems. Lecture Notes in Math. 1280. Springer-Verlag, Berlin 1987.

[18] Nomura, T. Manifold of primitive idempotents in a Jordan-Hilbert algebra. J. Math. Soc. Japan 45 1993, 37-58.

[19] Nomura, T. Grassmann manifold of a JH-algebra. Annals of Global Analysis and Geometry 12 1994, 237-260

[20] Upmeier, H. Symmetric Banach manifolds and Jordan C^{∗}-algebras, North Holland Math. Studies vol 104, Amsterdam 1985.