On Grassmannians associated with JB*-triples

Wilhelm Kaup ^{1}

Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen, Germany kaup@uni-tuebingen.de

Dedicated to Holger Petersson on the occasion of his 60^{th} birthday

1. An illustrating example

We illustrate our considerations in the following special situation: Let E: = C^{n×m} be the space of all complex n × m-matrices where 1 ≤ n ≤ m are fixed integers in the following. Every matrix z is considered as an operator C^{m} → C^{n} between finite-dimensional Hilbert spaces and ∥z∥ is the corresponding operator norm, i.e.

B = {z ∈ E : 11 − z^{∗}z > 0}

is the open unit ball of E where z^{∗} = \overline{z}′ is the transposed conjugate of z. It is known that the ball B is homogeneous under biholomorphic automorphisms and hence is an example of a bounded symmetric domain. Consider on E the linear transformation group

Γ: = \{ \begin{matrix}\{z ↦ azb or z ↦ az′b : a, b ∈ GL(n, C)\} & m = n > 1 \\ \{z ↦ azb : a ∈ GL(n, C), b ∈ GL(m, C)\} & otherwise. \end{matrix}

Then Γ is a complex Lie group that has precisely n + 1 orbits E_{0}, E_{1}, .., E_{n} in E – each E_{r} is the (locally closed) complex submanifold of all matrices of rank r. Clearly, E_{r−1} is in the closure of E_{r} for every 1 ≤ r ≤ n and E_{n} is the unique open orbit. For every pair of integers p, q ≥ 0 let 𝔾_{p,q} be the Grassmannian of all p-planes in C^{p+q}. Then 𝔾_{p,q} is a compact symmetric hermitian manifold of dimension pq holomorphically equivalent to 𝔾_{q,p}. Every E_{r} is in a canonical way a Γ-equivariant holomorphic fibre bundle over 𝔾_{m−r,r} × 𝔾_{r,n−r} with typical fibre GL(r, C) – simply associate to every z ∈ E_{r} the kernel and the range of z. Now consider the subgroup

G: = \{ \begin{matrix}\{z ↦ uzv or z ↦ uz′v : u, v ∈ U(n)\} & m = n > 1 \\ \{z ↦ uzv : u ∈ U(n), v ∈ U(m)\} & otherwise \end{matrix}

of Γ, where U(n) ⊂ GL(n, C) is the unitary group. Then G is a maximal compact subgroup of Γ and every g ∈ G is an isometry of E. In every E_{r} we pick a G-orbit S_{r} in a way that is motivated by our Jordan-algebraic approach in the general situation of arbitrary bounded symmetric domains: First consider the mapping c: E → E defined by c(z) = zz^{∗}z. Then c is a real-analytic homeomorphism of E, called the cube mapping on E. In a way, this mapping contains the full structural information of the space E. For instance, ∥z∥ < 1 holds for z ∈ E if and only if the sequence of iterates c^{k}(z) converges to zero for k → ∞. Let S ⊂ E be the fixed point set of c, that is the subset of all partial isometries e ∈ E (i.e. the restriction of e to (ker e)^{⊥} is isometric). S is a real analytic submanifold of E and every S_{r}: = S ∩ E_{r} is a G-orbit. In case r > 0 the manifold S_{r} is connected and contained in ∂B, and S_{n} is the Šilovboundary of the ball B. Clearly, S_{r} is a G-equivariant fibre bundle over 𝔾_{m−r,r} × 𝔾_{r,n−r} with typical fibre U(r). Also, there exists a G-equivariant real-analytic retraction ρ: E_{r} → S_{r} defined

1 Supported by a grant from the German-Israeli Foundation (GIF), I-0415-023.06/95.

by ρ(z) = lim_{k→∞} c^{−k}(z). A more direct way of describing this retraction is as follows: Every z ∈ E_{r} has a representation z = udv where d ∈ E_{r} is a (rectangular) diagonal matrix with nonnegative real entries (the singular value decomposition). Then ρ(z) = uρ(d)v with ρ(d) obtained from d by replacing all non-zero entries by 1. If instead we put z^{†} = ud^{†}v with d^{†} obtained from d by replacing every non-zero entry by its scalar inverse we get (up to a transposition) the Penrose inverse of z, and z ↦ z^{†} is a real analytic reflection of E_{r} about S_{r}. From

(1.1)  \overline{E}_{r} = ⋃_{j≤r} E_{j} = E_{0} + ∑_{j=1}^{r} E_{1}

it is easily derived that the group GL(E_{r}) does not depend on 1 ≤ r ≤ n. In particular, every g ∈ GL(E_{1}) induces a biholomorphic automorphism of M: = 𝔾_{m−1,1} × 𝔾_{1,n−1}. But M is the direct product IP_{m−1}×IP_{n−1} of complex projective spaces and all biholomorphic automorphisms of M are well known. This implies GL(E_{r}) = Γ for all 1 ≤ r ≤ n. A direct - but much longer - proof of this statement in the special case r = n = m is already contained in [23], compare also [10] and the references in the survey [1]. With the same complex analytic argument it can be shown that G is the group GL(D) of all linear isometries of the Banach space E.

The triple product structure on E = C^{n×m} can also be given by the family of all conjugatelinear operators Q(z) on E defined by Q(z)w = zw^{∗}z for all z, w ∈ E. Then a linear subspace I ⊂ E is called an inner ideal if Q(I)E ⊂ I – or equivalently – if IE^{∗}I ⊂ I holds. It can be shown that the inner ideals of E are precisely the spaces

I = {z ∈ E : z(H) = 0, z(E) ⊂ K}

for suitable linear subspaces H ⊂ C^{m}, K ⊂ C^{n}. Denote by 𝔾 the Grassmannian manifold of all linear subspaces of E. Then the linear group Γ ⊂ GL(E) acts holomorphically on 𝔾 and it is easy to see that every inner ideal of E has a closed Γ-orbit in 𝔾. On the other hand, a line L ⊂ E is an inner ideal of E if and only if its Γ-orbit is closed in 𝔾.

So far our discussion is only an illustration of what will be done in more generality in the following. Instead of a finite dimensional matrix space E will be an arbitrary complex Banach space for which the open unit ball B ⊂ E is a bounded symmetric domain. Banach spaces of this type (for instance all C*-algebras but also all complex Hilbert spaces) are called JB*-triples since they admit a natural Jordan triple product structure – or what amounts to the same – a canonical cube mapping c : z ↦ {zzz}. The elements of E can no longer be interpreted as operators, nevertheless we have the analogues of the groups G and Γ and also canonical fibrations over symmetric complex Banach manifolds. These are spaces of certain inner ideals in E and will be called the Grassmannians associated with the JB*-triple E. Our main result states that every connected component of the space of all complemented principal inner ideals of E is a symmetric complex Banach manifold.

Notation. For complex Banach spaces E, F, G we denote by L(E, F) the Banach space of all bounded linear operators E → F. Furthermore L(E): = L(E, E) is the Banach algebra of all continuous endomorphisms and E^{∗}: = L(E, C) is the dual of E. The group of all invertible operators in L(E) is denoted by GL(E). For every subset A ⊂ E we denote by GL(A) ⊂ GL(E) the subgroup of all g with g(A) = A. For every topological group G we denote by G^{0} the connected identity component of G.

By a sesqui-linear map L: E × F → G we always mean a mapping that is complex linear in the first and conjugate linear in the second variable. In particular, the inner product (u|v) of a complex Hilbert space is conjugate linear in v.

2. J*-triples and symmetric complex Banach manifolds

2.1 Definition. A complex Banach space E together with a continuous map

E × E × E → E, (x, y, z) ↦ {xyz}

is called a J*-triple (compare [17]), if for all a, b, x, y, z ∈ E the following conditions hold

(i) {xyz} is symmetric bilinear in the outer variables (x, z) and conjugate linear in the inner variable y.

(ii) {ab{xyz}} = {{abx}yz} − {x{bay}z} + {xy{abz}}.

(iii) The operator D(a, a) ∈ L(E) defined by c ↦ {aac} is hermitian, i.e. exp(itD(a, a)) ∈ GL(E) is an isometry for every t ∈ IR.

More generally, we denote for every x, y ∈ E by D(x, y) ∈ L(E) the operator z ↦ {xyz} and by Q(x) the antilinear endomorphism z ↦ {xzx}. Then Q(x) depends quadratically on x and satisfies the fundamental formula

Q(Q(x)y) = Q(x)Q(y)Q(x) for all x, y ∈ E .

The (not necessarily closed) linear subspace I ⊂ E is called a subtriple if Q(I)I ⊂ I holds and I is called an inner ideal if even Q(I)E ⊂ I is true. As a consequence of the fundamental formula for every x ∈ E the linear subspace Q(x)E is an inner ideal, called the principal inner ideal generated by x. For each non-empty subset A ⊂ E the set Ker(A): = {a ∈ E : Q(A)a = 0} is a closed inner ideal of E, called the kernel of A. A closed subtriple A ⊂ E is called complemented in E if E = A ⊕ Ker(A) holds, compare [7]. For instance, for every tripotent e ∈ E (that is {eee} = e), the principal ideal Q(e)E is closed and complemented in E.

We always denote by

G: = Aut(E): = {g ∈ GL(E) : g{xyz} = {(gx)(gy)(gz)} for all x, y, z ∈ E}

the automorphism group of E. Then G is a real algebraic subgroup of GL(E) in the sense of [11] and in particular is a real Banach Lie group in the norm topology of L(E). The corresponding Lie algebra is i Der(E), where

Der(E): = {δ ∈ L(E) : δ{xyz} = {(δx)yz} + {x(δy)z} + {xy(δz)} for all x, y, z ∈ E}

is the IR-linear subspace of all bounded derivations of E.

The J*-triples form a category in an obvious way – the morphisms are the bounded linear operators g satisfying g{xyz} = {(gx)(gy)(gz)} for all x, y, z in the domain of g. This category allows a duality in the following way: For a J*-triple E with triple product {xyz} let E^{d} be the Banach space E with triple product {xyz}^{d}: = −{xyz}. Then again E^{d} is a J*-triple, called the (triple) dual of E (to be distinguished from the Banach space dual E^{∗} of E).

J*-triples occur in the context of symmetric complex Banach manifolds, which are defined in the following way (compare [18] for details): Let M be a connected complex Banach manifold endowed with a lower semi-continuous norm ν: T M → IR on the tangent bundle T M that locally is equivalent to the norms given by local holomorphic charts. Then M is called symmetric if to every a ∈ M there is a (necessary unique) involutory biholomorphic isometry s_{a} of M with a as isolated fixed point. Then the group spanned by all symmetries s_{a}, a ∈ M, acts transitively on M. Choosing a point a ∈ M as base point, the tangent space E: = T_{a}M can be given the structure of a J*-triple in the following way: There exists an (essentially unique) local holomorphic E-valued coordinate z around a, such that every infinitesimal isometry of M is a polynomial of degree ≤ 2 in the local coordinate z. Furthermore, for every α ∈ E there is a unique bounded symmetric bilinear form q_{α}: E × E → E such that the vector field α − q_{α}(z, z) is an infinitesimal isometry of M. The triple product then is given by {zαw} = q_{α}(z, w). Changing the base point results in an isomorphic J*-triple. Every J*-triple occurs this way and every symmetric complex Banach manifold is uniquely determined by the corresponding J*-triple up to covering transformations.

3. JB*-triples

3.1 Definition. A J*-triple E is called a JB*-triple if for all a ∈ E

(i) D(a, a) has real spectrum ≥ 0,

(ii) ∥{aaa}∥ = ∥a∥^{3}.

Let E be a JB*-triple with open unit ball B in the following. Then it is known that ∥D(a, b)∥ ≤ ∥a∥·∥b∥ holds for all a, b ∈ E and also that the group G = Aut(E) coincides with the group GL(B) ⊂ GL(E) of all linear isometries of E. In case E is of finite dimension, the group G is compact. Besides G we also consider the structure group

Γ: = Str(E): = {g ∈ GL(E) : g{xyz} = {(gx)(g^{†}y)(gz)} for a suitable g^{†} ∈ GL(E)} .

It is known that g^{†} is uniquely determined by g ∈ Γ and that Γ ⊂ GL(E) is an algebraic complex subgroup, compare [18] proposition 5.6. In particular, Γ is a complex Banach Lie group in the norm topology of GL(E) and g ↦ g^{†} defines an antiholomorphic involutory group automorphism of Γ with G = {g ∈ Γ : g = g^{†}}. The Lie algebra of Γ is the closed linear subspace Der(E) ⊕ i Der(E) of GL(E). In particular, g: = exp (D(a, b)) ∈ Γ and g^{†} = exp (−D(b, a)) for all a, b ∈ E. For instance, if Ω is a locally compact topological space, the space E: = C_{0}(Ω) of all continuous C-valued functions on Ω vanishing at infinity is a JB*-triple with respect to the sup-norm and the triple product \{fgh\} = f\overline{h}g. Also, it can be shown that the corresponding structure group Γ consists of all linear transformations

f(ω) ↦ h(ω) · f(τ^{(}ω^{)})

where τ: Ω → Ω is a homeomorphism and h: Ω → C^{∗} is a continuous function with relative compact image.

3.2 Lemma. For every JB*-triple E and every a ∈ E the following conditions are equivalent. (i) a is (von Neumann) regular in E, i.e. a ∈ Q(a)E. (ii) a admits a pseudo inverse in E, i.e. there is a (necessarily unique) element a^{†} ∈ E with

a = Q(a)a^{†}, a^{†} = Q(a^{†})a and [Q(a), Q(a^{†})] = 0. (iii) e = lim_{n→∞} c^{−n}(a) ∈ E exists.

(iv) Q(a)E = Q(a)^{2}E.

(v) The inner ideal Q(a)E is closed and complemented in E. If these conditions are satisfied then e is a tripotent with D(a, a^{†}) = D(e, e) = D(a^{†}, a) that we denote by ρ(a). Also, Q(a^{†})Q(a) ∈ L(E) is a projection on E with range Q(a)E. Proof. The equivalence of (i) and (ii) is contained in [20], Lemma 4.1, which relies on results of [9]. Denote by F the smallest closed (complex) subtriple of E that contains a. Then there is a unique locally compact subset Ω ⊂ IR with Ω > 0 and Ω ∪ {0} compact together with a unique triple isomorphism φ: F → C_{0}(Ω) such that φ(a) = id_{Ω}. Furthermore, a is regular if and only if Ω is compact (compare [9], [20]), and then φ(a^{†}) = 1/φ(a). Furthermore, the sequence of all φ(c^{−n}^{(}a^{)}), n ≥ 1, converges locally uniformly on Ω to the function ≡ 1, which is in C_{0}(Ω) if and only if Ω is compact. This proves the equivalence of (i) and (iii) and also the second statement in (iii) is clear since for all b, c ∈ F the operator D(b, c) only depends on the product of functions φ(b)\overline{φ(c)} on Ω. Now assume (iv). Then a^{3}: = Q(a)a is in Q(a)^{3}E = Q(a^{3})E, i.e. a^{3} is regular and with it also a. On the other hand, if a is regular, then – as we will see later in this section – Q(a)E coincides with a Peirce-1-space and hence is closed and complemented in E. Finally, (iv) is a trivial consequence of (v).

Denote by E^{†} ⊂ E the subset of all regular elements in E. Then E^{†} = E holds if and only if E has finite rank. For instance, for E = C we have a^{†} = 1/\overline{a}, ρ(a) = a/|a| if a ≠ 0 and 0^{†} = ρ(0) = 0. More generally, if E = L(H, K) with complex Hilbert spaces H, K and triple product {xyx} = xy^{∗}x, then the operator a ∈ E is regular if and only if a has closed range in K and then a^{†} and ρ(a) are obtained in the following way: Write a as composition a = ȷp of the canonical mappings p: H → H/ ker(a) and ȷ: H/ ker(a) → K. Then

a^{†} = ȷ (ȷ^{∗}ȷ)^{−1}(pp^{∗})^{−1}p and ρ(a) = ȷ (ȷ^{∗}ȷ)^{−1/2}(pp^{∗})^{−1/2}p .

In particular, if a ∈ L(H, K) is invertible as operator we have a^{†} = (a^{−1})^{∗} and ρ(a) = a(a^{∗}a)^{−1/2}. Also, in an arbitrary JB*-triple E all invertible elements a (i.e. Q(a) is an invertible operator on E) are regular and a^{†} = Q(a)^{−1}a. For every pair of JB*-triples E, F and every triple homomorphism φ: E → F the inclusion φ(E^{†}) ⊂ F^{†} with φ(a^{†}) = φ(a)^{†} for all a ∈ E^{†} holds.

For every a ∈ E^{†} the space E becomes a complex Jordan algebra with respect to the product z∘w: = {za^{†}w}. Since a is an idempotent in this algebra we get a vector space decomposition (Peirce decomposition)

E = E_{1}(a) ⊕ E_{1/2}(a) ⊕ E_{0}(a)

where E_{k}(a) is the k-eigenspace of D(a, a^{†}). The Peirce spaces E_{1}(a) and E_{0}(a) are closed inner ideals, E_{1/2}(a) in general is only a closed subtriple of E. For every k denote by P_{k}(a) : E → E_{k}(a) the canonical projection (Peirce projection). Every P_{k}(a) is contractive and for k = 1/2 we have for instance

P_{1/2}(a) = 4 D(a, a^{†}) (11 − D(a, a^{†})) .

The structure group Γ leaves E^{†} invariant, actually g^{†}(a^{†}) = (ga)^{†} holds for all g ∈ Γ and all a ∈ E^{†}. In particular, g(E_{k}^{(}a^{)}) = E_{k}(g^{(}a^{)}) holds for all k and g, a as above. Consider a Γ-orbit N ⊂ E^{†}. Then N can be considered as a complex manifold immersed into E and for every a ∈ N we have

T_{a}N = E_{1}(a) ⊕ E_{1/2}(a)

for the tangent space at a to N.

Denote by S = Tri(E) the set of all tripotents in E, i.e. of all e ∈ E with {eee} = e. Then clearly S = {a ∈ E^{†} : a = a^{†}}. It can be seen that the element e: = ρ(a) of 3.2.iii is a tripotent with

D(a, a^{†}) = D(e, e) = D(a^{†}, a)

and hence

E_{k}(a) = E_{k}(e) = E_{k}(a^{†}) for all k and all a ∈ E^{†} .

Also, A(e): = {z ∈ E : z^{∗} = x}, z^{∗}: = Q(e)z, is a unital JB-algebra with E_{1}(e) = A(e) ⊕ iA(e). It is known that S is a real analytic submanifold of E and that the real (Banach) Lie group G = Aut(E) acts with open orbits on it. In particular, every connected component of S is a G^{0}-orbit. The tangent space to S at e is given by

T_{e}S = iA(e) ⊕ E_{1/2}(e) .

Two elements a, b ∈ E are called (triple) orthogonal in E if D(a, b) = 0, or equivalently, if D(b, a) = 0 holds. For every integer r ≥ 0 the JB*-triple E has rank ≥ r if there exist pairwise orthogonal, nonzero elements a_{1}, …, a_{r} in E. Also, for r ∈ \overline{IN}: = IN ∪ \{∞\} the JB*-triple E has rank r if it has rank ≥ r but not rank ≥ r + 1 in case r < ∞. The element a ∈ E has rank r in E if the closure of Q(a)E has rank r. Elements of rank 1 in E are called minimal or an atom in E. Also, an inner ideal I ⊂ E is called minimal (maximal, respectively) if it is a minimal (maximal, respectively) element in the partially ordered set (with respect to inclusion) of all inner ideals I of E with 0 ≠ I ≠ E. The inner ideal I is minimal if and only if dim I = 1 and hence if and only if I = Q(a)E for some atom a ∈ E, uniquely determined up to a complex factor ≠ 0.

Denote by ℙ the space of all complemented principal inner ideals and by 𝕄 ⊂ ℙ the space of all minimal inner ideals of E.

4. Grassmannians

Let E be a complex Banach space. Then a closed linear subspace X ⊂ E is called direct (in E) if there is a closed linear subspace Y ⊂ E with X ∩ Y = {0} and E = X + Y, i.e. E = X ⊕ Y as topological vector spaces. Every such Y is called a complement of X in E.

Denote by 𝔾 = 𝔾(E) the space of all direct linear subspaces of E, called the Grassmannian of E, compare [6] and [15] for details. It is known that 𝔾 has the structure of a complex Banach manifold uniquely determined by the following: For every X ∈ 𝔾 and every complement Y of X let U_{X,Y} ⊂ 𝔾 be the subset of all graphs G_{λ}: = {x+λx : x ∈ X} of linear operators λ ∈ L(X, Y). Then U_{X,Y} is an open coordinate neighbourhood of X in 𝔾 with holomorphic coordinate G_{λ} ↦ λ (called a canonical coordinate). The linear group GL(E) is a complex Banach Lie group acting holomorphically on 𝔾, i.e. the mapping GL(E)×𝔾 → 𝔾 defined by (g, X) ↦ g(X) is holomorphic. In the local coordinate neighbourhood U_{X,Y} the action of g ∈ 𝔾 has the following form: Represent g as an operator matrix (_{c d}^{a b}) with a ∈ L(Y), b ∈ L(X, Y), c ∈ L(Y, X) and d ∈ L(X). Then g is given by the linear fractional transformation λ ↦ (aλ + b)(cλ + d)^{−1}.

Now let E be a JB*-triple and denote by S = Tri(E) the subset of all tripotents. Furthermore, denote by 𝕁 ⊂ 𝔾 the subset of all direct inner ideals of E. It is easily verified that 𝕁 is a closed subset of 𝔾. As before, let Γ = Str(E) ⊂ GL(E) be the structure group and G = Aut(E) ⊂ Γ the automorphism group of the JB*-triple E. Then Γ acts holomorphically on 𝔾 leaving the subspaces 𝕁, ℙ and 𝕄 invariant.

Fix a tripotent e ∈ S in the following and put X: = E_{1}(e), Y: = E_{1/2}(e) and Z: = E_{0}(e). Then Y ⊕ Z is a complement for X in E. 4.1 Lemma. For every y ∈ Y and λ_{y}: = D(y, e) the operator g_{y}: = exp(λ_{y}) ∈ Γ maps X to

g_{y}(X) = {x + λ_{y}x + 1/2λ_{y}^{2}x : x ∈ X} ∈ ℙ

and the closed complex submanifold N: = {g_{y}(X) : y ∈ Y } of U_{X,Y ⊕Z} satisfies

N = U_{X,Y ⊕Z} ∩ 𝕁 .

Proof. Fix σ ∈ L(X, Y), τ ∈ L(X, Z) and suppose that the graph G_{λ} is an inner ideal for λ: = (σ, τ) ∈ L(X, Y ⊕ Z). Then {G_{λ}XG_{λ}} ⊂ G_{λ} is equivalent to

σ{uxv} = {(σu)xv} + {ux(σv)} and τ{uxv} = {(σu)x(σv)}

for all u, v, x ∈ X. The first equation implies for u = x = e the identity σ(v) = 2{(σe)ev}. For the proof of G_{λ} ∈ Γ(X) ⊂ ℙ we may therefore assume σ = 0. But then the second equation implies τ = 0, i.e. G_{λ} = X. 4.2 Corollary. ℙ is a closed complex submanifold of 𝔾 and every Γ-orbit in ℙ is open and closed in ℙ. For every tripotent e ∈ E and X: = E_{1}(e) the tangent space at X ∈ ℙ can be identified in a natural way with the Banach space E_{1/2}(e). In particular, we have a norm ν : T ℙ → IR on the tangent bundle T ℙ of ℙ. 4.3 Lemma. The natural action of G = Aut(E) on ℙ leaves the norm ν invariant, i.e. every g ∈ G is a biholomorphic ν-isometry of ℙ. For every connected component M of ℙ the real Banach Lie group G^{0} acts transitively on M. In particular, ν is continuous. Proof. Fix a tripotent e ∈ E and consider X: = E_{1}(e) ∈ ℙ. Identify Y: = E_{1/2}(e) with the tangent space T_{X}ℙ as above. For every y ∈ Y then h_{y}: = exp (2D(y, e) − 2D(e, y)) is in G^{0} and ψ(y): = h_{y}(X) defines a real analytic map ψ: Y → ℙ with differential dψ_{0} the identity on Y. By the implicit function theorem therefore the image ψ(Y) is a neighbourhood of X in ℙ and hence the orbit G^{0}(X) is open and closed in ℙ. The G-invariance of the norm ν follows from the fact that every g ∈ G is a linear isometry of E.

For a given tripotent e ∈ E consider the connected component M of ℙ that contains the point X: = E_{1}(e). For every t ∈ IR the operator γ_{t}: = exp ( − 2it D(e, e)) ∈ G^{0} fixes the point X ∈ M and the differential d_{X}γ_{t} is the operator e^{it} id on the tangent space T_{X}M = E_{1/2}(e). In particular, for t = π the transformation γ_{π} ∈ G^{0} (the Peirce reflection about E_{1/2}^{(}e^{)}) is a symmetry at X of M, i.e. M is a symmetric complex Banach manifold in the sense of [17]. Summing up we get the main result of this section. 4.4 Theorem. Let E be a JB*-triple. Then the space ℙ of all complemented inner ideals in E is a closed complex submanifold of the Grassmannian 𝔾 = 𝔾(E). Every connected component M of ℙ is a symmetric complex Banach manifold on which the group G^{0} acts transitively by biholomorphic isometries. For every X = E_{1}(e) ∈ M, e ∈ Tri(E), the tangent space T_{X}M can be canonically identified with the complex Banach space E_{1/2}(e). Furthermore, the (triple) dual of E_{1/2}(e) is the J*-triple associated with the symmetric complex Banach manifold M. Proof. Fix a tripotent e ∈ E and let M be the connected component of ℙ containing the point X: = E_{1}(e). It remains to show that the J*-triple associated with M is the (triple) dual of the JB*-triple Y: = E_{1/2}(e). For every y ∈ Y and a_{y}: = e + y + {yey} consider the Peirce space

X_{y}: = exp (2D(y, e))X = E_{1}(a_{y}) ∈ M .

Then U: = {X_{y} : y ∈ Y } is open in M and y ↦ X_{y} defines a biholomorphic map φ: Y → U. We identify U ⊂ M with Y via φ in the following. Denote by 𝔤 the real Banach Lie algebra of all holomorphic vector fields on M induced by the action of the group G on M. Then it is enough to show that for every α ∈ Y the holomorphic vector field α + {yαy} on Y ⊂ M belongs to 𝔤. For this fix α ∈ Y and denote by ξ ∈ 𝔤 the vector field induced by the one-parameter subgroup exp(IRη) of G, where

η: = 2D(α, e − 2D(e, α)) .

Since G consists of isometries of M and since 𝔤 contains the vector field y ↦ iy, ξ is a polynomial of degree ≤ 2, i.e. ξ = ξ_{0} + ξ_{1} + ξ_{2}, where ξ_{k} = ξ_{k}(y) is a homogeneous polynomial of degree k in y. Then

(∗)  \begin{aligned}ξ(y) + 2\{ξ(y) e y\} & = ξ_{0} + ξ_{1}(y) + 2\{ξ_{0} e y\} + ξ_{2}(y) + 2\{ξ_{1}(y) e y\} + 2\{ξ_{2}(y) e y\} \\ & ≡ η(a_{y}) mod X_{y} \end{aligned}

holds for all y ∈ Y as well as

(∗∗)  \begin{aligned}η(a_{y}) & = a − 2\{eαy\} + 2\{αey\} − 2\{eα\{yey\}\} \\ & = a − 2\{eαy\} + 2\{αey\} − 4\{\{eαy\}ey\} + \{yαy\} \\ & = α + 2\{αey\} + \{yαy\} + 4\{\{\{eαy\}ey\}ey\} \\ & − 2[\{eαy\} + 2\{\{eαy\}ey\} + 2\{\{\{eαy\}ey\}ey\}] \\ & ≡ α + 2\{αey\} + \{yαy\} + 4\{\{\{eαy\}ey\}ey\} mod X_{y} . \end{aligned}

For y = 0 the relations (∗) and (∗∗) imply (ξ_{0} − α) ∈ X ∩ Y, i.e. ξ_{0} = α and hence f(y) ∈ X_{y} for all y ∈ Y and

f(y): = ξ_{1}(y) + ξ_{2}(y) − {yαy} + 2{ξ_{1}(y) e y} + 2{ξ_{2}(y) e y} − 4{{{eαy}ey}ey} .

This implies that

exp ( − 2D(y, e))f(y) = ξ_{1}(y) + ξ_{2}(y) − {yαy} + higher order terms

is contained in X for all y ∈ Y. Therefore ξ_{1}(y) and ξ_{2}(y) − {yαy} are contained in X ∩ Y, proving ξ(y) = α + {yαy} for all y ∈ Y.

4.5 Lemma. Let e ∈ E be a minimal tripotent. Then the Peirce space Y: = E_{1/2}(e) is a JB*-triple of rank ≤ 2 and every JB*-triple of rank ≤ 2 occurs this way. Proof. Embed E in the canonical way into the bidual E^{∗∗} of E. Then also E^{∗∗} is a B*-triple with a separately w^{∗}-continuous triple product extending the one of E, compare [5] and [2]. The tripotent e is also minimal in E^{∗∗}, i.e. we may assume without loss of generality that E is a JBW*-triple. Now suppose that Y has rank ≥ 3. Then there exist pairwise orthogonal, non-zero tripotents u_{1}, u_{2}, u_{3} ∈ Y. Every non-zero tripotent u ∈ Y satisfies {uue} = λe for some λ ∈ C since e is minimal. λ = 0 would imply that e, u are orthogonal, i.e. only λ = 1/2 or λ = 1 is possible. But then λ ≥ 3/2 for the tripotent u: = u_{1} + u_{2} + u_{3} gives a contradiction. That every JB*-triple of rank ≤ 2 occurs in thnis way follows from the classification of all JB*-triples of finite rank in [19].

The question arises, to what extent the previous results extend from ℙ to the space 𝕁 or at least to the space of all Peirce inner ideals in the sense of [7]. For this, consider the following two special examples: First, suppose that E has rank 1, i.e. E = H for a complex Hilbert space H, say of dimension n. Then every linear subspace of E is an inner ideal. In particular, the full Grassmannian 𝔾(H) is the space of all closed inner ideals of E. Next, let E be a spin factor of dimension n ≥ 3, let the Hilbert space H with conjugation x ↦ \overline{x} and the corresponding symmetric bilinear form ⟨ | ⟩ on H be as defined in the next section. E and H coincide as topological vector spaces and a proper linear subspace I ⊂ E is an inner ideal if and only if I and \overline{I} are orthogonal in the Hilbert space H, or equivalently, if I is totally isotropic with respect to ⟨ | ⟩. For every pair p, q of cardinal numbers with n = 2p + q the space M_{p,q} of all closed inner ideals I with dim I = p and codim(I ⊕ \overline{I}) = q is a closed complex submanifold of 𝔾_{p,p+q} on which the orthogonal group O(H) (see the next section for its definition) acts transitively. But it is known that M_{p,q} has the structure of a symmetric complex Banach manifold only if p = 1 or q = 0, compare [16] for details.

5. The case of Cartan factors

In the following let H be a complex Hilbert space of dimension n with fixed conjugation x ↦ \overline{x}. Then ⟨x y⟩: = (x \overline{y}) defines a symmetric bilinear form H × H → C and

O(H): = {g ∈ GL(H) : ⟨x y⟩ = ⟨gx gy⟩ for all x, y ∈ H}

is the (complex) orthogonal group of H. The unitary group of H will be denoted by U(H). The real form X: = \{x ∈ H : \overline{x} = x\} of H is a real Hilbert space and O(X): = U(H) ∩ O(H) is the (real) orthogonal group.

For every pair p, q of cardinal numbers with p + q = n we denote by 𝔾_{p,q} the space of all closed linear subspaces of H with dimension p and codimension q. Then 𝔾_{p,q} is a connected, simply connected, symmetric complex Banach manifold on which U(H) is a transitive group of holomorphic isometries. The corresponding J*-triple is the (triple) dual of the JB*-triple L(X, Y), where X ∈ 𝔾_{p,q} is arbitrary and Y: = X^{⊥} is the orthogonal complement of X in H. Furthermore, 𝔾_{p,q} is biholomorphically equivalent to 𝔾_{q,p} and IP_{n−1}: = 𝔾_{1,n−1} is the projective space of H.

Now let K be a further complex Hilbert space with fixed conjugation also written as y ↦ \overline{y}. Denote by m the dimension of K and assume for easier notation that H = K in case n = m and also that both conjugations then coincide. For every z ∈ L(H, K) we get operators \overline{z} ∈ L(H, K) and z′ ∈ L(K, H) uniquely determined by \overline{z}(\overline{x}) = \overline{zx} and ⟨zx y⟩ = ⟨x z′y⟩ for all x ∈ H and y ∈ K. Clearly, z^{∗} = \overline{z}′ holds and z ↦ \overline{z} defines a conjugation on the Banach space L(H, K).

An important class of JBW*-triples is given by the Cartan factors, compare for instance [19] p. 473. These come in six types I–VI and will always be denoted by E in the following. As before, let Γ be the structure group and let 𝕄 be the space of all minimal ideals in E. Since every Cartan factor is irreducible, i.e. not a direct sum of non-zero ideals, 𝕄 is connected and hence a symmetric complex Banach manifold of rank ≤ 2 as a consequence of Theorem 4.4 and Lemma 4.5. The complex Banach Lie group Γ acts holomorphically and transitively on 𝕄, let Φ: Γ → Aut(𝕄)) be the corresponding group homomorphism. 5.1 Lemma. The kernel of Φ is the center {λ id : λ ∈ C^{∗}} of Γ. Proof. Fix g ∈ ker Φ. Then by assumption there is a function f: 𝕄 → C^{∗} with g(a) = f(L_{a}) · γ(a) for all atoms a ∈ E, where L_{a}: = Ca is the corresponding point in 𝕄. It is easily seen that f is holomorphic. But on 𝕄 every holomorphic function is constant: In case E has finite dimension this follows from the fact that then 𝕄 is compact. In case E has finite rank this follows from the fact that then any two points in 𝕄 are contained in a compact connected complex submanifold of finite dimension. In the case of infinite rank it follows from the fact that then 𝕄 is a product of one or two projective spaces (types I and III) or a Grassmannian 𝔾_{2,n−2} (type II). Therefore we may assume f ≡ 1 and hence that g fixes all atoms in E. This implies that the operator g^{†} has the same property. Now let x ∈ E be arbitrary. Then {axa} = g^{†}{axa} = {a g(x) a} for every atom a ∈ E implies g(x) = x and finally g = id.

The following is a list of the different types of Cartan factors. Γ: = Str(E) denotes always the structure group of the Cartan factor E. I_{n,m}: E: = L(H, K). The transposition z ↦ z′ defines a triple isomorphism L(H, K) → L(K, H), i.e. we may assume n ≤ m. For every cardinal number r ≤ n let E_{r} be the set of all operators a ∈ E with closed range of dimension r in K (E_{r} should not be confused with Peirce spaces in E). Then E^{†} is the union of all such E_{r} and every E_{r} is invariant under the structure group Γ. For r finite E_{r} is connected and coincides with the set of all elements of (triple) rank r. In particular, for r = 1, the associated space 𝕄 of minimal inner ideals is biholomorphically equivalent to IP_{m−1}×IP_{n−1} and as a consequence of Lemma 5.1 it is easily derived that (compare also [22] for a different approach)

Γ = \{ \begin{matrix}\{z ↦ azb or z ↦ az′b : a, b ∈ GL(H)\} & m = n > 1 \\ \{z ↦ azb : a ∈ GL(K), b ∈ GL(H)\} & otherwise. \end{matrix}

For g ∈ Γ given by z ↦ azb then g^{†} ∈ Γ is given by z ↦ (a^{∗})^{−1}z(b^{∗})^{−1}. All Γ-orbits in E_{r} are connected and are obtained in the following way: For every triple of cardinals p, r, q with p + r = n and r + q = m put

E_{p,r,q}: = {a ∈ E_{r} : dim (ker a) = p, dim (coker a) = q}

where coker a = K/a(H) is the cokernel of a. Then the associated symmetric manifold of inner ideals can be identified with 𝔾_{p,r} × 𝔾_{r,q} and the canonical fibration

E_{p,r,q} → 𝔾_{p,r} × 𝔾_{r,q}

is given by a ↦ (ker a, im a).

II_{n}: E: = {z ∈ L(H) : z′ = −z}. For every cardinal number r with 2r ≤ n let E_{r} be the set of all operators a ∈ E with closed range of dimension 2r in H. Then E^{†} is the union of all such E_{r} and every E_{r} is Γ-invariant. For r = 1, the associated space 𝕄 of minimal inner ideals is biholomorphically equivalent to 𝔾_{2,n−2}. On 𝔾_{2,n−2} the group GL(H) acts in a natural way as a group of collineations, and by [15] this gives the full group Aut(𝔾_{2,n−2}) of biholomorphic automorphisms in case n ≠ 4 (for n = 4 there is a second connected component consisting of all correlations with respect to non-degenerate symmetric bilinear forms on C^{4}). As a consequence we have

Γ = {z ↦ aza′ : a ∈ GL(H)} if n ≠ 4 .

On the other hand, type II_{4} will also occur below as type IV_{6}. All Γ-orbits in E_{r} are connected and obtained as follows: For every pair of cardinals p, r with p + 2r = n put

E_{r,p}: = {a ∈ E_{r} : dim (ker a) = dim (coker a) = p} .

Then the associated symmetric manifold of inner ideals can be identified with 𝔾_{2r,p} and the canonical fibration

E_{r,p} → 𝔾_{2r,p}

is given by a ↦ im a.

III_{n}: E: = {z ∈ L(H) : z′ = z}. For every cardinal number r with r ≤ n let E_{r} be the set of all operators a ∈ E with closed range of dimension r in K. Then E^{†} is the union of all such E_{r} and every E_{r} is Γ-invariant. For r = 1, the associated space 𝕄 of minimal inner ideals is biholomorphically equivalent to IP_{n−1} and hence

Γ = {z ↦ aza′ : a ∈ GL(H)} .

All Γ-orbits in E_{r} are connected and obtained as follows: For every pair of cardinals p, r with p + r = n put

E_{r,p}: = {a ∈ E_{r} : dim (ker a) = dim (coker a) = p} .

Then the associated symmetric manifold of inner ideals can be identified with 𝔾_{r,p} and the canonical fibration

E_{r,p} → 𝔾_{r,p}

is given by a ↦ im a.

IV_{n}, n ≥ 3: E is the Hilbert space H endowed with the equivalent norm

∥z∥ = \sqrt{(z|z) + \sqrt{(z|z)^{2} − |⟨z, z⟩|^{2}}}

and the triple product

\{xyz\} = ⟨x, \overline{y}⟩z + ⟨z, \overline{y}⟩x − ⟨x, z⟩\overline{y} .

E is called the complex spin factor of dimension n. There is a unique principal inner ideal of rank 2 (the whole space itself). The element z ∈ E has rank ≤ 1 if and only if ⟨z, z⟩ = 0. Therefore, 𝕄 can be identified with the quadric ℚ_{n−2}: = {Cz ∈ IP_{n−1} : ⟨z, z⟩ = 0} of dimension n − 2. By [16]. p. 161 Corollary, every biholomorphic automorphism of ℚ_{n−2} is induced by a linear transformation g ∈ O(H). On the other hand, every g ∈ O(H) in contained in the structure group Γ since g^{†} = \overline{g}. Together with Lemma 5.1 this implies

Γ = C^{∗}· O(H): = {z ↦ λ · g(z) : λ ∈ C^{∗}, g ∈ O(H)} .

V: E is the exceptional JB*-triple of dimension 16, compare [21]. E has rank 2, and as a consequence of Theorem 4.4 the space 𝕄 is biholomorphically equivalent to the quadric ℚ_{10}. The space of rank 2 principal inner ideals is biholomorphically equivalent to the quadric ℚ_{8}.

VI: E is the exceptional JB*-triple of dimension 27. The space 𝕄 ⊂ IP_{26} of minimal inner ideals as well as the space of rank 2 principal inner ideals is biholomorphically equivalent to the compact exceptional symmetric hermitian space of dimension 16 with associated J*-triple the (triple) dual of V.

Let E be a Cartan factor and denote by F ⊂ E the closure of the set of all finite rank elements in E. Then F is a JB*-subtriple of E, called an elementary Cartan factor. E, F have the same rank and for the types I – III F is the set of all compact operators in E, whereas for the types IV – VI F coincides with E. In any case, F is the smallest non-zero norm-closed ideal in E. Furthermore, F is w^{∗}-dense in E and E can be identified with the bidual of F in a natural way. For every integer r ≥ 0 with r ≤ rank(E) denote, as before, by E_{r} ⊂ F the set of all elements of rank r in E. The norm closure \overline{E}_{r} satisfies the identity (1.1) and this implies immediately for every λ ∈ GL(E).

5.2 Lemma. λ(E_{r}) = E_{r} if and only if λ(E_{1}) = E_{1}. Finally we have by the above reasoning: 5.3 Proposition. Let E be a classical Cartan factor (i.e. of type I – IV) and let r ≤ rank(E) be a positive integer. Then

(i) The structure group of E consists of all w^{∗}-continuous operators λ ∈ GL(E) with λ(E_{r}) = E_{r}.

(ii) The structure group of the elementary Cartan factor F consists of all operators λ ∈ GL(F)

with λ(E_{r}) = E_{r}. (iii) The restriction operator λ ↦ λ|F induces an isomorphism of structure groups Str(E) →

Str(F).

For a JB*-triple E denote by E^{−1} the set of all invertible elements a ∈ E, i.e. Q(a) is invertible as operator on E. Then E^{−1} is contained in E^{†} and Q(a^{†}) = Q(a)^{−1} for all a ∈ E^{−1}. Every g ∈ Γ: = Str(E) leaves E^{−1} invariant as a consequence of

g ∘ Q(a) = Q(g^{(}a^{)}) ∘ g^{†} .

E^{−1} is open in E but may be empty. For every a ∈ E^{−1} the mapping E → E defined by z ↦ exp(D(a, z))a has real derivative Q(a) at z = 0, i.e. the orbit Γ^{0}(a) is a neighbourhood of a by the implicit function theorem. This implies that Γ^{0} acts transitively on every connected component of E^{−1}.

Denote by U = U(E) the set of all unitary elements of a ∈ E, i.e. D(a, a) = id. Then

U(E) = E^{−1} ∩ Tri(E) = {a ∈ E : Q(a)^{2} = id}

is easily verified and also that the group G: = Aut(E) leaves U invariant. It is known that U is a closed real analytic submanifold of E and that G^{0} acts transitively on every connected component of U. The mapping ρ: E^{−1} → U defined in Lemma 3.2 has connected fibres. Therefore, E^{−1} and U have the same number of connected components.

Now fix an element e ∈ E^{−1}. Then E becomes a Banach Jordan algebra J with unit e by defining x ∘ y: = {xe^{†}y}. Every g ∈ Γ with g(e) = e satisfies g^{†}(e^{†}) = e^{†} and hence g(x ∘ y) = g(x) ∘ g(y). Also, g is isospectral in the sense of σ(g^{(}a^{)}) = σ(a) for all a ∈ E and spectrum defined by σ(a): = {λ ∈ C : λe − a ∉ E^{−1}}.

The Cartan factors containing invertible elements are precisely the types I_{n,n}, II_{2n}, III_{n}, IV_{n} and VI with n an arbitrary cardinal number. For E of type I – III the element a ∈ E is invertible (unitary) if and only if a is invertible (unitary) as Hilbert space operator. For E of type IV and VI, invertibility is equivalent to having rank 2 and 3 respectively. U and hence E^{−1} are connected, in particular, Γ acts transitively on E^{−1}.

For the rest of the section let E be of type I_{n,n}, i.e. E = L(H) with dim H = n, and denote by e the identity operator on H. Then E^{−1} = GL(H) and by a result of [14] every surjective isospectral linear operator g: H → H is of the form z ↦ aza^{−1} or z ↦ az′a^{−1} for some a ∈ GL(H). But clearly, the surjective linear operator g: H → H is isospectral if and only if g(e) = e and g(GL(H)) = GL(H). Since Γ acts transitively on GL(H) we get another characterization of the structure group. 5.4 Lemma. Γ consists of all surjective linear operators g: H → H with g(E^{−1}) = E^{−1}.

References

1. Aupetit, B.: Sur les transformations qui conservent le spectre. preprint

2. Barton, T. J., Timoney, R. M.: Weak*-continuity of Jordan triple products and applications, Math. Scand. 59, 177-191 (1986)

3. Braun, H., Koecher, M.: Jordan-Algebren. Berlin-Heidelberg-New York: Springer 1966 4. Chu, C.H., Isidro J.M.: Manifolds of tripotents in JB*-triples. Math. Z., to appear

5. Dineen, S. Complete holomorphic vector fields on the second dual of a Banach space. Math. Scand. 59, 131-42 (1986)

6. Douady, A.: Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné. Ann. Inst. Fourier 16, 1-95 (1966)

7. Edwards, C.M., Rüttimann G.T.: Peirce Inner Ideals in Jordan*-Triples. J. of Algebra 180, 41-66 (1996)

8. Faulkner, J.R.: On the Geometry of Inner Ideals. J. of Algebra 26, 1-9 (1973)

9. Fernández, A., Garcia, E., Sánchez, E., Siles, M.: Strong regularity and generalized inverses in Jordan systems. Communications in Algebra. 20, 1917-36. (1992)

10. Frobenius, G.: Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, I. Sitzungsberichte Königl. Preuss. Akad. Wiss. 994-1015 (1897)

11. Harris, L.A., Kaup, W.: Linear algebraic groups in infinite dimensions. Ill. J. Math. 21, 666-674 (1977)

12. Hanche-Olsen, H., Størmer, E.: Jordan operator algebras, Monographs Stud. Math. 21 Pitman, Boston-London-Melbourne 1984

13. Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press 1962

14. Jafarian, A.A., Sourour, A.R.: Spectrum-Preserving Linear Maps, J. of Funct. Analysis 66, 255-261 (1986)

15. Kaup, W.: Über die Automorphismen Graßmannscher Mannigfaltigkeiten unendlicher Dimension, Math. Z. 144, 75-96 (1975)

16. Kaup, W.: On the Automorphisms of Certain Symmetric Complex Manifolds of Infinite Dimensions. Anais da Academia Brasileira de Ciências 48, 153-163 (1976)

17. Kaup, W.: Algebraic Characterization of Symmetric Complex Banach Manifolds. Math. Ann. 228, 39-64 (1977)

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

19. Kaup, W.: Über die Klassifikation der symmetrischen Hermiteschen Mannigfaltigkeiten unendlicher Dimension I, II, Math. Ann. 257, 463-483 (1981); 262, 503-529 (1983)

20. Kaup, W.: On spectral and singular values in JB*-triples, Proc. Royal Ir. Acad. 96A, 95-103 (1996)

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

22. Lutscher, F.: Cartanfaktoren, reelle Formen und ihre Strukturgruppen. Diplomarbeit Universität Tübingen 1997

23. Marcus, M., Moyls, B.N.: Linear transformations on algebras of matrices. Can. J. Math. 11, 61-66